What is the difference between Beal's Conjecture and Fermat's Last Theorem? In Beal's Conjecture, are we assuming that we are allowed to repeat numbers? The conjecture is the following: 

Beals Conjecture: If $a^n+b^n = c^n $where $n \geq 2$ and $a,b,c,n \in \mathbb{Z}^{+}$ then $a,b$ and $c$ have a common prime factor.

But isn't this basically the opposite of Fermat's Last Theorem? Or does Fermat's Last Theorem require $a,b,c$ to be distinct?

This is not Beal's conjecture.  You have in fact stated Fermat's Last Theorem.  Beal's conjecture states that if $a^x + b^y = c^z$, where $a,b,c > 0$ and $x,y,z > 2$, then $\gcd(a,b,c) \neq 1$.  Fermat's Last Theorem is an easy corollary of Beal's Conjecture.
