The main reason to keep them separate is for modularity. It's easier to deal with interactions and bugs between the two programs when they're separate and specialized, plus as mentioned before you can use your lobby program to download/patch the real game program.

If you put them both into the same program, you're essentially writing 3 programs: one for your lobby, one for your game, and one to join them together into a single application. This makes things more complex and prone to bugs.

Actually, the main reason to keep them separate on Windows is that a running executable cannot overwrite its own file. Even on Linux, doing so results in undefined behavior, although you can get around this with renames -- something you also can't do to a running executable on Windows.
So, the patcher can update the main game EXE before it's started. Once the game is started and the patcher has exited, the game can in turn update the patcher if needed.

That certainly makes more sense from a patch standpoint, but the OP was discussing the benefits of a separate lobby program that doesn't necessarily have to be a patcher; it's just one of the other things that it could do if it were separated from the main game program. :>

Updates and patches are also possible from within the game if the developers aren't too lazy.
Sure you can't overwrite the executable and you shouldn't overwrite the resource files while the game is running. But take a look at Diablo 2 it already had a patch downloader build into the actually game. When finished the game starts a new process (the Blizzard updater) and quits. The Updater does it's thing and then starts the updated game.

Back to he topic, when using a separate lobby application.

• It can be easier to build complex gui's, tables and so on. You can use a Gui library that you don't want or can't use in the actual game. E.g. Qt but maybe you don't want your whole game based on Qt. Maybe you prefer to use IM gui in the game.
  
• You can even replace the lobby with a webpage and somehow launch your game directly from the site. This enables you to use html, javascript, databases without having to write your own customize master server and you can take advantage of all the new web technologies (remember to check your targeted system requirements). And you only need an updater/patches for your actual in-game part of the game, the rest can be updated on the fly (test before putting it on your production system!).
  
• You can make the lobby application look and feel like every other native application for that OS. Which make it look more like it was developed for that OS. Not some fancy widgets or the look of another operating system. On Windows it looks like Windows, on OS X like OS X and on Linux like Linux (is that even possible with all the different desktop managers?).
  
• Bad side, if the whole thing is done wrong it feels like some loosely coupled applications that barely work together.

I really like the Gui of the old Unreal Tournament (the really old) where the developers put a whole desktop in the game. Nowadays they have switched too a more console friendly version which is not exactly great to use with mouse and keyboard.

Edit: Something more on the patcher... If you have a small skeleton application that is loading your game as a library, you can unload that library, overwrite it and the load it again. (Beware, I believe unloading is not possible on OS X!)

FYI, "UFO: Enemy Unknown" used a separate (DOS) exe file for the two main parts of the game - the strategic and tactical elements. As far as I know, it used temporary files or something similar, to store the relevant data which was transferred between them - DOS did not allow multitasking, so one had to quit (or at least pause and free as much memory as possible) to start the other.