TITLE: Character tables and simple groups.
QUESTION [8 upvotes]: Is it known if there are two finite non-isomorpic non-abelian simple groups with the same character table?  Does this answer change if the subsidiary information (like the orders and sizes of the conjugacy classes) is included?  (There are no examples of this in the "ATLAS")

REPLY [14 votes]: There are no such two non-isomorphic groups. As is pointed out in Neil Strickland's answer, up to some exceptions, a finite simple group is determined by its order. For these exceptions it is known that the smallest character degree larger than 1 is different 
(For the infinite series $B_n(q)=O_{2n+1}(q)$ vs $C_n(q)=PSp_{2n}(q)$ for $q$ odd this follows from the results in Landazuri and Seitz (J. Algebra 32 (1974), 418–443, MR0360852 (50 #13299)).
This means that a finite simple group is determined in the class of finite simple groups by its character degrees with multiplicities, and led Huppert (Illinois J. Math. 44, 4 (2000), 828-842, MR1804317 (2001k:20009)) to the conjecture:  

Conjecture: If two finite groups $G$ and $H$ have the same set of character degrees (without counting multiplicities) and $G$ is nonabelian simple, then $H\cong G\times A$ for some abelian $A$.  

This has been verified for some simple groups, but is still open to the best of my knowledge. As mentioned in comments, Tong-Viet (MR2905242) has shown that finite simple groups are determined by their character degrees with multiplicities among all finite groups. Needless to say that all this depends heavily on the classification.