The Latin word traces to the Proto-Indo-European root *gʷet- "pitch"; see that link for other cognates.
The expression "bitumen" originated in the Sanskrit, where we find the words "jatu", meaning "pitch", and "jatu-krit", meaning "pitch creating", "pitch producing" (referring to coniferous or resinous trees). The Latin equivalent is claimed by some to be originally "gwitu-men" (pertaining to pitch), and by others, "pixtumens" (exuding or bubbling pitch), which was subsequently shortened to "bitumen", thence passing via French into English. From the same root is derived the Anglo Saxon word "cwidu" (Mastix), the German word "Kitt" (cement or mastic) and the old Norse word "kvada".
The word "ašphalt" is claimed to have been derived from the Accadian term "asphaltu" or "sphallo," meaning "to split." It was later adopted by the Homeric Greeks in the form of the adjective ἄσφαλἤς, ἐς  signifying "firm," "stable," "secure," and the corresponding verb ἄσφαλίξω, ίσω meaning "to make firm or stable," "to secure".
The word "asphalt" is derived from the late Middle English, in turn from French asphalte, based on Late Latin asphalton, asphaltum, which is the latinisation of the Greek  (ásphaltos, ásphalton), a word meaning "asphalt/bitumen/pitch", which perhaps derives from , "not, without", i.e. the alpha privative, and  (sphallein), "to cause to fall, baffle, (in passive) err, (in passive) be balked of".
The first use of asphalt by the ancients was as a cement to secure or join various objects, and it thus seems likely that the name itself was expressive of this application. Specifically, Herodotus mentioned that bitumen was brought to Babylon to build its gigantic fortification wall.
From the Greek, the word passed into late Latin, and thence into French (asphalte) and English ("asphaltum" and "asphalt"). In French, the term asphalte is used for naturally occurring asphalt-soaked limestone deposits, and for specialised manufactured products with fewer voids or greater bitumen content than the "asphaltic concrete" used to pave roads.