They have quite the similar meaning, a slight difference can be found if you think as "chiedere" as a general translation for "ask" (ex. Chiedere di passare il sale - ask to pass the salt) while domandare suits better the translation for "ask" when it's specifically a question (domandare dove si trova la banca - ask where is the bank). This is a very slight difference though as most of the time the two verbs are interchangeables :)