The alto-clarinet and the modern form of the bassett horn are looking very similar. They are low clarinets (in Eb or in F respectively). Today they have the characteristic bend in the barrel which is usually made from metal - and they usually show a metal funnel like a bass clarinet although some do not; it seems that is more an esthetical feature. Due to the bends the clarinet can be built roughly the same hight as a Bb clarinet and it is usually played just held with your hands (and optionally with a neck strap), while bass clarinets do have a thorn to be placed on the ground (for holding, a wooden bass clarinet is too heavy).
You find alto clarinets in harmony bands or symphonic bands, hardly ever in classical symphony orchestras. And there was only one classical composer who wrote for bassett horn at all; but this composer was Wolfgang Amadeus Mozart and the bassett horn was his favorite instrument. Therefore alone it is very likely that this instrument will stay with us forever.
The famous clarinet concerto in A was - as far as we know today - originally composed for a bassett clarinet and not for the A clarinet. That instrument might have looked a little like the Bassetthorn, maybe without the funnel. Since such instruments don't exist any more, the musicians use an A clarinet and changed some of the lowest notes to be able to play that parts. There are original hand-written notes by Mozart and since he precisely knew the instruments and their tone range we can tell that something like this must have existed.
Today there still are some very old bassett horns in museums (e.g. in Hamburg and Berlin - see picture) and they all do have the charakteristic bend and a box-like connecting piece (the "book") - and with some the bell looks like a french horn. This is probably the reason why it was named bassett "horn". The crooked form was due to the technical problems building long and still effective keys. The player had to reach and cover nearly all tone holes with his fingers - in Mozart's days the modern key was not yet invented.
The modern Bassett Clarinet is less a Bassett Horn but more an extended version of an A- or B-flat Soprano clarinet. Today professional clarinet players (like Sharon Kam and Sabine Meyer) use this type of instrument to play Mozart's famous concerto KV 622 in its original form.