Apparently the racers have different characteristics. Do they have different speeds, accelerations, handling etc. or can I choose the racer I want? (tiptup)
They have different stats it seems. Though those look like the N64 version stats (it has banjo). If you're on the DS the stats might be different.
Edit: Here is the stats for the DS version.
Edit 2: To more directly answer your question, you can choose whichever you like, but you should be aware of the stats if you'd like to have a better chance of winning. Both links list situations in which each racer does best (and the wiki has a few cons as well).
Yes, their stats are different. You can find information on them easily online, such as this GameFAQ. It's for the DS version, but also applies to the N64.
The starting characters are pretty similar. IIRC Tiptup and Timber were the overall best. Sometimes I would use Krunch as his extra speed really helps, especially for boss races.
Special characters are much much better however.
Taj/Wizpig are by far the best if you can control them. Before that (and on N64) use Drumstick and T.T.
Although, it also depends on the track, for instance Taj/Wizpig are terrible on tracks that require precise turns (including bonus minigame tracks).