The old answer based on the books can be found further down. But as GreenMat insists on an answer from the movies I have to say: There is NO logical explanation in the movies for what happens to Voldemort. Neither does the disintegration happen in any other instance of a duel nor is there any spell in the movies that can cause something like this.
The closest thing to the disintegration we get is the effect on Quirrell when touching Harry in the first movie, where something very similar happens.
So a movie- explanation MIGHT be, that it is the protection from his mother that lets the bad guys disintegrate. But still this is inconsistent with the fourth movie, where Voldemort -after getting his new body- can touch Harry without disintegrating.
As you can see in 4th, 5th 6th and 7th movie: whenever Voldemort tries to kill Harry he uses the unforgivable curse number one Avada Kedavra.
It always looks the same (green flash) as the spell in the last duel. And Harry always uses Expelliarmus, the red beam, especially in the duel in the fourth movie at the graveyard.
We also see in the movies that Voldemort is able to do magic without actually saying the words (e.g. when he forces Harry to bow by just pointing his wand at him.
These two facts let us assume, that it was Avada Kedavra again -this time unspoken- in the last duel.
WHY the bounced back spell lets Voldemort disintegrate while normally it "just kills" is not explained but could be made to refer to Quirrell in the first movie.
Answer from the book:
The movies are quite inaccurate in a lot of ways, and this is one of them.
In the books there is no difference between casting a spell and duelling. In the 6th book we learn, that experienced wizards can cast spells by just "thinking" them without speaking them out loud. They learn the "unspoken spells" at school.
It is said in the books that casting a spell without saying it loud gives you an advantage especially in duels, as your opponent cannot prepare for the spell as he doesn't hear it.
In the final duel in the book none of the two uses an unspoken spell: Harry casts his favorite spell "Expelliarmus" while Voldemort uses -guess- "Avada Kedavra". As Harry wins Voldemort's spell bounces back to him and kills him (this is NOT normal in a duel but due to the fact of the Elder Wand's true allegiance).
And no, he does NOT disintegrate, this is made for the effect of it only in the movie.
To answer your question: the loser just experiences the spell the winner cast. Of course a Death Eater might kill the loser with another spell after winning, but that is not part of the duel.