dimmi = di' a me = tell me (please)
-- "tell me anything" = DIMMI tutto
dirmi = dire a me = about telling me:
-- "I asked him to tell me anything" = Gli ho chiesto di DIRMI tutto
Please note that in "dimmi" the real voice of the verb "dici" becomes di' in the exhortative form and dropping the last syllable causes the apostrophe, which should not be confused with the stress of "dì" which means "morning" (buondì = good morning).