I am of the belief that computers have played a very minor role in modern mathematics compared to it influence in science, social science, technology, and culture.
Fundamentally, the benefits of computers is speed, efficiency, and accuracy in computation and calculation. Computers can do a millions of computation in the time it take us to do a single one. (Just imagine how much computation is needed to load this webpage.) You can imagine how much science and technology have advance since people can now do things and obtain knowledge millions of times faster.
The above describes how computers have been useful in many areas. It is because people knew how to solve the problem and realized the problem consists of absurdly large amount of individually simple tasks. Hence a computer can be used to quickly and accurate preform these computation, and obtain results in an instant rather than years.
In mathematics, there are known examples of theorems proven using computers to analyze thousands of cases. This is because the theorem has been reduced to many relatively simple and computational tasks. However, I believe that for most of the important research in abstract mathematics the difficultly lies not in the prodigious amount of sheer computation, but rather an understanding of the problem. Mathematician do not yet understand the problem to be able to divide the question in many, many individually simple tasks for a computer to calculate. Moreover, many open questions in mathematics are more conceptual and do not fit the computational nature suitable for computer calculations.
There are probably areas of mathematics that have greatly benefitted and have been revolutionized by computers; however, there is no ubiquitious presence of computers in mathematics as seen in many of the other ares of science and social science.
The above discussed the computers mostly used as a computation aid. There are some work in using computers to generate or assist in the proof of theorems. But they have not been used to the extent that they have "abolished" the traditional mathematicians proof.
Somewhat relevant to computers, mathematicians have recently begun to study computability itself and the computable aspect of areas of mathematics. Some of this likely preceded the computers. These would include the study of algorithms, Turing Machines, Oracles Machines, computable functions, the Turing Degrees, complexity theory, effective randomness, computable model theory, etc.