I think the real major difference (this is technically a response to the two above posters) is the torque amount.
The 908 has the most torque of all LMP/Group C cars as far as I can remember, while the 905 has one of the lowest amounts of torque. The 905 also is lighter by 70 kilos if I remember correctly, which could account for what Hyped is saying.
Thing is... I've got the 908 on my main, and the 905 on my alt. In a comparison test I did (before update 2.09 - I need to redo this test) using a standard G25 racing wheel, I was about 5 seconds faster in the 905 on Sarthe than the 908. Both on Racing Hards.
I couldn't tell you what the exact setup was, but I DID put them both at 1000 kilos for an even test to see who was faster with restrictions. If I remember right, the 905 was very sharp through the turns - even when I slammed the gas, it didn't kick out or slide whatsoever. This, I believe, is due to the lower torque. (Only driving aid on was ABS - set to 5 for both.)
The 908, however, was all over the place before I got my tires warmed up (about five laps before I felt comfortable and went for some hot laps) and I do believe that this is due to the 908's higher torque curve compared to the 905. They both did handle almost the same once the tires were warmed up enough, though.
That being said, I've run both cars in the 24 Hours of LeMans pre-2.09, and the 908 was better on fuel economy and tire wear than the 905. Why? I really don't know. I drove with the same style throughout the race, which was a calm pace because I had lapped the field due to them having to pit a lot more frequently than I did.
It could have also been that me and 2 friends rotated driving duties just like real LeMans drivers do - 8 hour stints, me with the G25 and them with DFGT's. It could theoretically made a difference. The only actual way to test which one is better overall is to identical conditions and everything else it - which isn't possible at the moment, none of us can reproduce what our lines exactly were and how we took each corner.
If you got fed up with how long and dully informative of a post this was, I'll give you my opinion; The 908 may be better for enduros, but the 905 is a lot more fun to drive.