What is the difference between Rudin's *Principles of Mathematical Analysis* and *Real and Complex Analysis* books?

It doesn't make much sense to compare the two books, as they (for the most part) cover different material. The bulk of Principles of Mathematical Analysis is devoted to a rigorous introduction to single variable and multivariable calculus. In contrast, Real and Complex Analysis covers measure theory, some functional analysis and Fourier analysis, and complex analysis.

Based on your description of what you want, I'd say you should start with Principles of Mathematical Analysis.

As for the style of the books, Rudin is famous for his "slick" proofs and difficult exercises. Some people like his books a great deal, and some don't. It might be a good idea to take a look at them (if possible) before buying one.

Principles, the so-called baby Rudin, is undergraduate introduction to analysis. Basic proof writing, continuity, derivative, integral. It has some advanced topics which normally aren't covered in an undergraduate course such as a rigorous introduction to differential forms and a crash course on measure theory. It was meant to be covered in a two semester sequence by mathematics undergraduates. The downside of the book is the lack of any pictures. The story I heard was that Walter had lots and lots of pictures, but at the time the publisher said they'd print no pictures as it would be too expensive to typeset, so they all got removed.

The other book, sometimes called big Rudin is graduate level analysis. It combines what are usually the graduate level real and complex analysis classes that phd students take in their first or second year.