what is the difference in usage of 'pertain' and 'appertain'?

Solution 1:

Pertain conveys an association with--at least in part. Appertain conveys a total association with.

Solution 2:

There is considerable overlap in meaning, but the most striking feature is that pertain seems to be far more popular. Records for pertain in the Corpus of Contemporary American English outnumber those for appertain by 120 to 1. The contrast in usage in British English is nowhere nearly as pronounced, but the British National Corpus still shows a preference for pertain by almost 3 to 1. The preference is confirmed, if not as decisively, by citations in the Oxford English Dictionary, where those containing pertain are more than one and a half times greater in number than those containing appertain.