Many people in the mod community were very excited to see the release of the Unreal Developer Kit (UDK) a few months ago.  Along with generating excitement into a very dedicated community, it also introduced many new modders into a flourishing area of indie-development.  However, since UDK is free, most beginners jump right into UDK, which is OK though you might just benefit more from purchasing a shelf-copy of Unreal Tournament 3.

UDK
UDK is a free full version of UnrealEd (the editor environment used to create games like Gears of War 1/2, Bioshock 1/2, and of course Unreal Tournament 3).  The editor gives you all the features of the editor from the shelf-copy of the game plus some refinements in many of the tools.  (One of the first things you'll find about UnrealEd is that it's a collection of tools grouped into the same editor so it really isn't a single 'tool')

Interestingly enough, Epic is allowing you to sell any game made in UDK with a few catches.  First off, you must purchase a liscense for your game (which, I THINK is aproximately $99 starting).  Secondly, you must pay 25% of all profits for the first $5,000 of your game revenue to them (about $1250).  Finally, you cannot use any of the 'media' provided in UDK for your game.  UDK provides sample meshes, textures, materials, sounds, and other sample pieces of media pulled (mostly) from Unreal Tournament 3.

The final point here will really determine whether you should use UDK.  There is a very small amount of media provided in UDK for someone to go in and begin creating levels without first developing your own meshes, textures, and other media.  Sure, you can slap together a few unique levels, though you will end up finding yourself restriced to the same items over and over and over.  This is absolutely how professional game development is; you are 'given' (typically liscensed or built in-house) an engine/editor and you begin creating all the content for the game and placing it.  UDK is aimed toward those who really want to build their game content from scratch with a currently existing engine.  It is not suited for someone who would like to simply build levels and quick mods without learning external 3D programs and image editing software.

Unreal Tournament 3
Unless you have a serious grudge against FPS's, Epic, or your computer sucks, there really is no reason not to own this game for PC.  You can pick it up on Steam or Amazon for around $20 brand new.  Not only are you provided with a full single-player and multiplayer game, but you are given the entire UnrealEd 3.0 including all of the content used to build UT3.  If you want to start building levels and mods quickly for UT3, you should absolutely pick up a shelf-copy.

However, as off-the-shelf UT3 is a few years old now, the tools have not been updated for quite a while.  Compared to UDK, the menus are more difficult to navigate through and take more time getting used to.  Since UDK is updated almost every month, there are new inclusions to the editor that may not be in UT3 (including the future addition of 3D!).  I haven't worked enough with shelf UT3 to see if there are more features in UDK or if they both feature the same stuff in different forms, however you should remember that the Unreal Engine 3.0 has undergone numerous upgrades between it's launch and Gears of War 2 (in fact, Epic had a conference to show off what changed just between the Gears of Wars games).

Since UT3 has much more core content, someone who wants to focus on level editing or modding the core UT3 game may find their needs better suited with an off-the-shelf copy of UT3.  If that level designer has a team that is generating custom assets, they may be better off with UDK.

The choice is now yours...