Papers from different authors with similar results submitted to the same conference Me and my coauthors have recently submitted a paper to conference X. By talking to colleagues I discovered that another set of authors have had a very similar result, and submitted their paper to conference X.

Basically, the result is a theorem that solves an open problem stated in a paper which appeared at conference X in the previous year. The new theorem is essentially the same in both papers, but the proofs are very different. 

What happens in this case?
This sounds like a straight-forward case of multiple discovery.  

In the case that both papers are well-written and have correct proofs, the typical and appropriate thing would be for both of them to be accepted.  If one or both are not sufficient for the standards of the conference in their presentation or technical approach, however, then it would be reasonable for one or both to be rejected, however, even if they have identical results.

Given that there is often a great degree of variability in reviewing, however, it would likely be a good thing to get in contact with the conference organizers and let them know about the situation, so that they can ensure fair treatment of the two papers (e.g., via meta-review or by assigning the same reviewers to both).