Is it known if there are two finite nonisomorpic nonabelian 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")

There are no such two nonisomorphic 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), 828842, MR1804317 (2001k:20009)) to the conjecture:
This has been verified for some simple groups, but is still open to the best of my knowledge. As mentioned in comments, TongViet (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. 


The order of $G$ is the sum of the squares of the character degrees, so is determined by the character table. According to this question: a finite simple group is determined up to isomorphism by its order, with just a couple of exceptions. 


I believe the answer to the basic question here is that no one expects to find two such nonisomorphic groups. However, even with the classification of finite simple groups (probably) in hand, the question is not easy to resolve. There are however a number of papers showing that specific types of finite simple groups are characterized uniquely by their character tables. Here are random samples of the older literature from MathSciNet: MR0316551 (47 #5098), Lambert, P. J., Characterizing groups by their character tables. I. Quart. J. Math. Oxford Ser. (2) 23 (1972), 427–433. MR0404415 (53 #8217), Pahlings, H., Characterization of groups by their character tables. I, II. Comm. Algebra 4 (1976), no. 2, 111–153; ibid. 4 (1976), no. 2, 155–178. I'm not at all an expert on these developments in finite group theory, but it's important to recognize the difficulty of the underlying problem. ADDED: Neil Strickland has pointed to a very useful earlier discussion on MO, which indicates that the question is indeed resolvable if the classification is known. The earlier work I sampled tried to deal with characterizations by character tables without using the classification, in some cases successfully. The groups of Reetype were certainly a big obstacle for some years. 

