Tagalog is the native language of the Tagalog people of Central Luzon which has dialects such as Bulaceño, Manila, Batangueño, etc.
Filipino is the national language based on the grammar and structure of Tagalog. It also aims to incorporate from other Philippine languages such as Bisaya and Ilocano. Vocabulary-wise, it is also heavily influenced by Spanish and English.
Example:
Filipino: bintana (ventana), silya (silla), mesa (mesa), kotse (coche)
Tagalog: dungawan, upuan, hapag, sasakyan