Zip is a very old compression system. It has a number of weaknesses compared to modern algorithms including 'toy-grade' encryption, relatively poor compression, and poor tolerance to file corruption. Zip, however, has the benefit of great age - support for it is outstanding, with many compression/decompression programs available for every OS in use.
RAR was designed from the ground-up to be extensible and is now considerably more advanced than Zip, which has hardly changed since the late 1980s. Features that have been added include industry-grade encryption using public algorithms, full support for massive files, advanced error recovery, multiple optimised compression algorithms with automatic selection, etc. The price you pay for this, is that there are multiple versions of the software, with some features not necessarily backwards compatible.
Still, it could be worse - several different software publishers have extended Zip in similar ways to provide large file support, and enhanced encryption. The trouble is their approaches are incompatible!
Personally, I use RAR. WinRAR is quick and easy to use. The 'profiles' function in WinRAR is great for backups - I just stick a DVD-RW in the drive, double click the 'backup my photos' icon on the desktop, and WinRAR automatically updates the archive on DVD-RW. Error correction means that even if the disc develops a few unreadable sectors I am unlikely to lose any data.