Technically speaking, a papal bull is a format for a document from the Roman Chancery. They are written in the third person, refer to the pope as "Bishop, [and] Servant of the Servants of God", and are affixed with the papal seal in lead (known as a bulla in Latin, from which the term "bull" is derived). It is very time-consuming to produce these types of documents, so the Roman Chancery has developed other formats (e.g., the papal brief) for regular use, reserving the bull format for only the most solemn of documents. The papal decree appointing a new bishop to a diocese, for example, is still regularly issued as a papal bull.

Encyclicals (more properly, encyclical letters), on the other hand, are a type of document. They are the weightiest form of papal letter, generally used to communicate a substantial teaching on a significant topic. The term "encyclical" is originally Greek and means "circulating", in the sense that an encyclical letter is not private, but to be publicly circulated among its intended recipients. Most encyclicals are issued as briefs rather than bulls.

As for the relative frequency of bulls and encyclical letters, I don't have any hard numbers to compare, but I suspect that the increasing ease with which encyclical letters can be published, copied, and distributed has led to their increasing popularity with the Holy See. Papal bulls, on the other hand, have declined in use because (as previously mentioned) they are quite costly and time-consuming to prepare properly.