Both are action RPGs but Borderlands leans more towards the action side, whereas Fallout New Vegas has more role-playing elements. One major difference is that Fallout NV lacks any type of multiplayer.
Vast, vast difference. I love both, but Fallout is a non-linear RPG first and foremost that lets you just wander around, discover people/places/missions. It's very emersive and can be heavily modded (custom user mods that change/augment the game). It's very similar to the Daggerfall series (Oblivion being the latest in the series) where you can more or less make the gaming experience whatever you want it to be and you can follow (or not) the main storyline as you choose.
Borderlands is a fun shooter first and foremost. It's mostly linear and has a good sense of humour and is very light on the RPG elements. It's pretty much the same experience for everyone with little variation other than some special skills that are unique to 1 of 4 characters.
Not to mention Fallout's graphics blow Borderlands away. The whole experience from FO is better in my opinion. Borderlands was a good game but the DLC's were not that fun. I'm playing Old World Blues now for FO and it's great. The others are right about which one is more RPG and Action/shooter.
I think Borderlands is a cheap, shallow rip-off of Fallout. Borderlands has far less depth than Fallout. Fallout 3 and New Vegas are vast, complex, deep games from a content, gameplay, and development point of view. They are awesome achievements when you think about how much work and complexity goes into creating the fallout games. Borderlands is much less complex and has had much less effort put into developing a rich gameplay experience. In other words, Borderlands is a cheap game. They didn't develop the gameplay enough.