This summer I owned both at the same time, and preferred the Basic 9 due to handle and blade shape. SethMurdoc has already covered most of the differences; I'll just add a couple points. First, the handle swell on the Basic is slightly less drastic than on the Rat, and I found the softer Res C more comfortable on the Basic as well. But by far my favorite thing about the Basic 9 was the blade. Though the Rat and the B9 have nearly the same length of blade measured from the handle, the small choil on the Basic means you have quite a bit more edge. This, combined with the narrower blade SethMurdoc mentioned, makes for a wicked-looking knife. Looks don't have any effect on performance, but I found myself smiling every time I unsheathed the Basic 9.
I was nerdy enough to take some measurements when I had both.
Basic 9 Battle Rat
OAL 15 14 1/2
Blade length 10 9 7/16
Handle length 5 1/16 5 1/16
Choil width 9/16 1 1/16
Edged blade 8 15/16 7 7/16
Blade width 1 5/8 1 13/16
Grind width 1 5/8 1 1/4
EDIT- the spaces disappear when I submit the post.

Not sure how to unjumble it.
I did this with a long ruler, so error is about +- 1/16. Thats why the Basic's blade + handle doesn't exactly equal its OAL.
From the numbers you can see the Basic 9 has a narrower, longer blade with more edge. The Battle Rat has a choil you can put your finger in to choke up on the blade.
As mentioned, the steels are different. The tempering of the blades is also different: the Basic 9 is through-tempered (i.e. the same hardness throughout), while the Battle Rat is differentially tempered (the edge of the blade is at full hardness, while the spine is softer. This makes the blade less likely to break under lateral load (than a through-hardened knife of the same geometry, heat treat, and steel)). The Battle Rat also has the Busse penetrator tip, while the Basic does not.
The Battle Rat is still in production, while the Basic 9 was discontinued years ago. Though both have the same warranty, if you somehow manage to monkey up your Basic 9, the Busse shop won't be able to replace it with another Basic 9.
The Battle Rat is less expensive. The cheapest I've seen a Basic 9 sell for here was mine: $150.00. Most are sold for $180-200. The Battle Rat is around $150.00 plus shipping from Swamp Rat.
The early production Basic 9's had a smooth black epoxy coating (like mine) and black handles; later production had the tougher black crinkle coat with black handles. The Battle Rat is available in black crinkle coat w/black handle or in desert tan crinkle coat w/tan handle. The desert crinkle coat is supposedly more durable than the black.
This is all I can think of (I'll exclude sheaths from the discussion because a wide array of custom sheaths are available for both, and I'm getting tired of typing

).
Jeremy