The Team Fortress 2 Beta game is where the devs test changes to the game prior to rolling them out officially. It isn't intended to replace the "real" game for normal play; the rules for acquiring items are very different, for example, to facilitate testing.
More detailed information is available from the official wiki.
If you want to play on the same PC:
Yes. Steam should recognise that the game is already installed. Failing that (which definitely works on games newer to Steam, but I'm unsure about older games like TF2), copy the contents of your SteamApps folder to his. (see the SteamApps\username instructions below)
If you want to play on seperate PCs:
Yes. Open your SteamApps folder. This will be depend on your operating system and be:
C:\Program Files\Steam\SteamApps
for 32 bit (older) Windows systems,
C:\Program Files (x86)\Steam\SteamApps
for 64 bit (newer) Windows systems, and
/Users/<username>/Library/Application Support/Steam/SteamApps
for Mac OS X systems.
You will need to copy all files who's name begins with "source" or "team fortress 2" from that folder on your PC to the folder on your brother's PC. There is also a file called winui.gcf which you will need to copy in there also.
Then follow the instructions in the below heading.
SteamApps\username folder instructions
You will also need to copy the <your steam username>\team fortress 2 folder inside your SteamApps folder to SteamApps\<your brothers steam username>\team fortress 2. If there are any folders beginning with source or winui in there as well, you should copy those also. Apart from required files for the game itself, these folders contain configuration and downloaded maps.
Notes
If you miss any individual files then they will be downloaded when Steam launches the game.
These instructions will not work for Windows to Mac, or Mac to Windows.
Best Answer
Try using the "verify integrity of game cache" feature in Steam. Should be able to find that if you right click your game, go to properties, and look under all of the tabs (if I remember right the last tab). If it still does not work try deleting the local content and re-downloading the game. P.S. Sometimes the simple solution is the best solution. Hopefully that works :D