The first answer above by NReilingh covers the physical appearance of the instruments but there is also an acoustic difference.
A Slide trumpet has a large bore of .460-.470" and is played using a trumpet mouthpiece. It also has a long tapered lead-pipe inside the upper tube just like any other trumpet.
A Soprano trombone has a more trombone like smaller bore of around .420-.430" and the better ones use a smaller shank mouthpiece like a modern cornet. That mouthpiece should have a deeper cup and larger throat than a trumpet mouthpiece. It will have either no tapered lead-pipe or a short one with little taper as it is with trombones.
That said, if the modern trumpet did not change and evolve to it's present large bore version but stayed smaller bore with little to no lead-pipe, like they were in the 19th century, then there would be no real difference between the two in acoustics.
The sound of the two instruments is not the same. The trumpet is brighter and blasts louder and harsher then the trombone.
I play both instruments but usually prefer the Soprano trombone over the Slide trumpet both for ease of playing and sound.
It's a shame that many trumpet makers market their Slide trumpets as mini trombones or something like that. They are in fact clouding the issue. Most of the time they are quite ignorant that there is a difference.
By the way, the Eb sopranino trombones do not play any higher than the Bb soprano, so they do not contribute other than the novelty of them. There is also a very cute Bb Piccolo trombone.