I sampled a couple of notes from a Bass Clarinet, played by the amazing Linda.
A middle C on a Bass clarinet has a very similar shape to the bottom notes of a "standard" Bb Clarinet. Which makes perfect sense, since the Bass Clarinet is, in fact, a clarinet.
Here's the low E from a standard clarinet.
Middle C of the Bass Clarinet. The shape is almost identical, with a few extra harmonics peeking through (the ones at C#, F# ish, and A# ish). So there's a definite difference, but it is subtle. The Bass Clarinet, is a little more complex in tone, as you'd expect, since that's you want one in the texture of your music.
Bass Clarinet Low register has many more harmonics that add to the richness of the sound. It still sounds like a clarinet, however there's a lot more harmonics that we can hear because the note is so much lower.