Asher,
Many good answers have been provided by Nicolas, Bart and Jerome. I would like to attempt at making the information a bit more structured, if I may.
A comparison between PS and LR in general is a bit like comparing apples and oranges. Although both programs have large overlapping functionalities, they also have certain ones which the other does not have. Therefore a comparison is only meaningful within each functional area. So let's make a high level list of those:
1) Digital Asset Management (DAM)
2) Raw conversion
3) Image editing
4) Printing
5) Exporting
6) Color management
7) Add-ons
We now can compare LR and PS for each of these areas.
1) LR is a fully featured DAM application. It provides all the necessary features to ingest images and tag, keyword and categorize them during ingestion. After ingestion, one can review images and rate and select them easily using flags, color labels, keywords, smart collections, etc. One can create regular and smart collections (based on dynamic selection criteria), stack images, etc. One can also export collections to web pages, books, etc. Also, starting with LR5, one can create portable catalogs of images without having to carry the original source images, using the smart cache function. All this is done within LR using a uniform user interface and customizable keyboard shortcuts, which makes it very efficient and easy to use. LR is a database driven application, all image information is kept in this database, such as keywords, exit data, parametric edits, etc.
PS has some of the DAM features of LR via Bridge, such as ingesting, tagging, keywording, labelling, etc. But it does not have collections or off-line catalogs. Export facilities are rather limited. Bridge is not database driven like LR, it has to operate on a folder structure.
So in the area of DAM, there is no discussion that LR wins from Bridge. One can decide to use LR for DAM purposes and not for image editing, which can be done in PS or any other image editing program. So LR does have an added value when used as a DAM application, regardless whether one has PS or not.
2) Raw conversion. Both LR and PS use the same ACR engine for converting raw files into rastered images. The functionality of both is the same although the gui of LR is a bit nicer. As long as they are at the same version level, there is no difference in the resulting images. So this is a draw between LR and PS.
3) Image editing is the area where the bulk of the comparison is to be made.
Firstly, the modus operandi of LR is parametric editing, meaning that all edit actions are defined by the values of the editing parameters. The editing parameters are predefined in LR, such as exposure, contrast, shadows, highlights, clarity, vibrance, HSL, noise reduction, sharpening, etc. One edits the image by changing the values of these parameters (e.g. by shifting the exposure slider to +1 or by clicking the lens correction checkbox). The resulting changes are not physically applied to the original image, which remains unchanged at all times. The changed image which is shown to the user is a rendered intermediate file (which is called a preview file). Whenever the parameters change, LR creates a new preview file according to the newest parameter settings. LR saves all these changes (i.e. the parameter values) in its database. It can also save them externally a sidecar file (xmp) or embed them in the original tiff, jpg and dng files without changing the original image.
One important aspect of real parametric editing is that the sequence of changing the parameters is not important. For example, changing the sharpening first and then exposure will result in the very same image if one changes the exposure first and then the sharpening. As a side note, this no longer applies to LR5 where one can use both the upright function and the clone brush. In that case, it is important to do the upright function first and only afterwards to the cloning.
So LR is a parametric, non-destructive editing program, whereas PS is not, at least not at the level of LR. And therin lies the rub. Although many LR fans will use this as the main argument to prove its superiority, the fact is that one can achieve similar levels of parametrization within PS by using adjustment layers and smart filters. If one uses layers for all the changes made to the image, one can achieve non destructive editing. But it has its downsides. Firstly, the history of changes cannot be saved with the image nor applied to other images automatically (such as copying and pasting or synchronizing of the development settings in LR). Additional layers (unless they are adjustment layers) add to the size of the image. Also, the work done in PS has to be saved as a file using one of the many available formats such as psd, tif or jpg (can be hundreds of megabytes), whereas the work done in LR is saved in the LR database as parameters, taking only a few kilobytes of space. For example: a raw file of 25 MB, when processed with LR, will result in a 25MB raw file (unchanged), a few KB size increase in the LR database, a 2-3 MB preview file and a few MB ACR cache file. The same raw file, when processed in PS using 16-bit RGB, will result in a 25 MB raw file (unchanged), a 100-300 MB tif/psd file and a few MB ACR cache file (the same as above). If one has a photo collection of let's say 10,000 raw files out of which 2,500 are to be eventually edited , then the extra disk space required for the PS users would be around 400 GB. For someone like myself who has 200K+ images, the difference becomes a staggering 8 TB. This adds a lot of complications and cost for keeping the master files and backing them up properly. Therefore, if the majority of one's images are meant for displaying on the web galleries or forums, one would be better off using the LR as a parametric editor from which web versions can be exported rather than using PS as the only editing program. I know that I can get superior results for my images by using the full blown functionality of PS and its associated add-ons. But the fact is, most pictures I share here in OPF do not deserve that kind of editing nor the resulting huge master files. A quick parametric editing in LR is usually enough to have a representative version of the picture good enough for web. LR offers the convenience of doing this with just a few clicks and exporting jpg files using presets. As such, it is essential to my workflow even though I also have PS.
One of the most important advantages of PS is of course the possibilty to work with layers and layer masks as well as the blending modes. That opens up endless options to edit one's image in any way one wants. LR has a primitive version of layers too, which can be achieved using adjustment brushes and gradients. They are similar to straightforward adjustment layers of PS using a normal blending mode (with opacity adjustments). The difference with PS is that the adjustment brushes are also parametric. Their layer sequence does not have an influence on the end result, whereas in PS the sequence of layers is rather important and can change the end results. For example, let's say that in LR, one paints with an adjustment brush on the image and sets the exposure to +1.0. After that, one paints another adjustment brush on the same image area using an exposure of -0.4. Then the image area will end up having a total exposure value of +0.6, which is an accumulation of the all individual parameter settings in LR. As you can imagine, PS works differently. It would increase the exposure +1.0 in the first layer and then decrease it -0.4 in the second one. But the end result would be more like -0.2 exposure, because the change is based on a previous change and the sequence is important. Also, where PScan have a totally different image in each of the layers, LR adjustment brushes (as considered to be quasi layers) all act upon the single base image.
Anyway, we can safely conclude that the layering functionality of PS wins hands down from LR. Especially if one utilizes the added value of blend modes and blend if functions, which make a huge difference.
PS has also a lot of additional functions which are non-existent in LR, such as typesetting, paths, selections, etc. Many of these are not applicable to straight forward photo editing, but one cannot live without the selection and cloning facilities of PS when one needs to "photoshop" the image such as removing unwanted elements or cloning certain areas in a smart way. Yes, LR has also a limited version of cloning but the total functionality is way limited compared to PS. So even if it is one in hundred pictures which may need such sophisticated cloning/extraction/etc, the photographer still needs to use PS (or other editing programs)occasionally. What do the photographers do who only have LR then? Well, they just shoot other images which do not require the extensive photoshopping.
What about some other functions such as stitching panoramas or creating HDR images? Well, these are the areas where PS offers built in functionality, albeit not top of the line compared to 3rd party specialized solutions. If one uses programs such as autopano, ptgui of pt assembler for stitching, PS does not offer any added value compared to LR. The same applies to HDR programs such as SNS-HDR or photomatix. PS is valuable if one does not use any of these task specific applications.
So when it comes down to the question, if one could have either LR or PS for image editing purposes, which one should one have, the answer would have to be PS considering the 100% functional coverage of all image editing situations. Of course this is not taking into account the total package of functionalities in other areas such as DAM nor the cost implications. To be honest, can a photographer survive with LR only without PS? Yes, definitely, although every once in a while one will have to trash a picture which could have been salvaged in PS.
4) Printing. The printing module of LR is easier to use and more functional (layout possibilites) compared to PS. LR also uses the option to print at the native print resolution of the printer used and it applies some smart output sharpening. LR is thus a better option compared to PS for printing images.
5) Exporting. Within PS, one can save an image in various formats. On top of that, bridge can be scripted to export multiple images in various formats as well as upload an publish them to web sites such as facebook or flickr. LR can do the same and even a bit more. I like the possiblity of using plug-ins in LR which can replace certain export functions with better algorithms. Such as using the mogrify plug-in to use Lanczos or Sync methods in downsampling the exported images and not the inferior bicubic used by adobe, My vote goes to LR in this area.
6) Color management. Although both applications are fully color managed, the flexibility of PS is of course industry standard. LR has only one working color space (melinda rgb) whereas the user can choose any working color space in PS. Also, PS can be configured for a variety of output color spaces, especially CMYK ones for publishing. LR has recently gained the soft proofing possibility. Before that, one had to do a round trip to PS to soft proof any output files. Anyway, for regular photographers, the color management of LR is fully adequate. For anybody who does publishing, such as Nicolas, PS would be an essential application in the process of creating their output files.
7) Add-ons. Both PS and LR have endless plug-ins and filters which enhance the functionality of both applications. Most major commercial add-on producers (such as Nik or Topaz) make their solutions available for both platforms. LR used to be a bit lagging compared to PS but in the recent years any new add-on which comes out has to be created for both apps to be commercially viable. One has to realize that using a plug-in such as Nik's ColorEfex in LR creates a tiff/jpg version of the raw file (with or without the parametric editing already done on the raw image). So it breaks the parametric editing workflow paradigm. This is an area which would be a draw between LR and PS.
So finally, the answer you have been waiting for Asher. Why would one need LR if one has PS? For the reasons of having a real DAM system, a better streamlined and easier workflow, more efficient publishing to web and the like, easier and better printing and saving considerable hard disk space by having parametric editing.
I hope that this helps.