Latent Dirichlet Allocation vs Hierarchical Dirichlet Process
HDP is an extension of LDA, designed to address the case where the number of mixture components (the number of "topics" in document-modeling terms) is not known a priori.  So that's the reason why there's a difference.

Using LDA for document modeling, one treats each "topic" as a distribution of words in some known vocabulary.  For each document a mixture of topics is drawn from a Dirichlet distribution, and then each word in the document is an independent draw from that mixture (that is, selecting a topic and then using it to generate a word).

For HDP (applied to document modeling), one also uses a Dirichlet process to capture the uncertainty in the number of topics.  So a common base distribution is selected which represents the countably-infinite set of possible topics for the corpus, and then the finite distribution of topics for each document is sampled from this base distribution.

As far as pros and cons, HDP has the advantage that the maximum number of topics can be unbounded and learned from the data rather than specified in advance.  I suppose though it is more complicated to implement, and unnecessary in the case where a bounded number of topics is acceptable.