Regular asymptotic analysis looks at the performance of an individual operation asymptotically, as a function of the size of the problem. The O() notation is what indicates an asymptotic analysis.
Amortized analysis (which is also an asymptotic analysis) looks at the total performance of multiple operations on a shared datastructure.
The difference is, amortized analysis typically proves that the total computation required for M operations has a better performance guarantee than M times the worst case for the individual operation.
For example, an individual operation on a splay tree of size N can take up to O(N) time. However, a sequence of M operations on a tree of size N is bounded by O( M(1+log N) + N log N ) time, which is roughly O(log N) per operation. However, note that an amortized analysis is much stricter than an "average-case" analysis: it proves that any possible sequence of operations will satisfy its asymptotic worst case.
