They responded in both similar and different ways. Both resisted colonial settlement and rule, sometimes violently, always persistently. The Buddhists resisted in both nations, but were more important in the Burmese resistance to colonialism by the British. The Brits also owned India at the time, and made Burma a province of that colony, so they had to deal with being subcolonized by India too.
Three separate Anglo-Burmese wars were fought in the 19th century in attempts at independence.
Later they were able to negotiate their independence from Great Britain about the same time that India did.
Vietnam was colonized by the French, and resistance there finally became serious about the time before World War II. Unlike Burma, the communists in Vietnam were more organized and more numerous. They sided with the Alies in World War II in hopes of throwing out the Japanese and getting independence, only to be recolonized by the French. They fought the French and defeated them in a nine year war. Next it was America's turn. So unlike Burma, who resisted over a long period of time, Vietnam's rebellions occurred in a concentrated period from 1940 - 1975. Also unlike Burma, they won their Independence through force.