1) RAR files are far tighter compression than ZIP. Typically a ZIP archive decompressed and recompressed as RAR will be about 10-15% smaller. For some filetypes such as word documents and excel spreadsheets, the RAR version is often less than half the size
2) Solid compression lets you further this gap.
3) Recovery blocks in whatever quantity you want so that the archive has redundancy within it in case some disk blocks go bad or a downloaded segmented archive is missing/corrupted. (Although SmartPAR is really better for this).
4) Special non-lossy multimedia compression algorithms can be applied to bitmaps, audio and text (although text isn't really multimedia....) Sometimes the multimedia compression gets a better ratio than JPEG with ZERO QUALITY LOSS (unlike JPEG).
5) WinRAR has a superior interface
6) Handles ZIP files faster than WinZIP does.
Seriously, the only reasons ZIP is still around are:
1) Built into Windows XP
2) Better marketing to software companies with tools to have executable files run while they are ZIP compressed internally.
3) Better multi-platform support (StuffIt RAR plugin on mac doesn't support full featureset and won't decompress any archives with special options like solid archiving turned on).