When is the best time of the year to visit Zanzibar?

Weather wise the best time is late June to Early March.
Late March to Mid June is considered the rainy season although this is not the monsoon type scenario you hear from some people. In April you get an average of 10 or 11 days rain in the month. It does not rain the whole day and night but it can come down rather convincingly at times.
==============
zanzibar airport