In American English, the verb burgle, meaning to rob, is regarded as a humorous backformation from burglar, and burglarize is the preferred term in serious contexts.
In British English, it’s the other way around. Burgle is a legitimate verb, used even in sober news reports, and burglarize (or burglarise, as it would probably be spelled if it were an accepted word in British English) is virtually nonexistent in serious contexts. Some Britons view burglarize as an American barbarism.
Irish, Australian, New Zealand, and South African writers tend to go along with British writers on this. Canadians prefer burglarize.
Burglar has a long history going back at least to the Medieval Latin burglator and probably beyond. Burgle and burglarize both came about in the late 19th century—neither is significantly older than the other—developing separately on opposite sides of the Atlantic.
For example, these major American publications prefer burglarize in serious contexts:
They are accused of entering vehicles parked on the 1900 block of Muirfield to burglarize them. [Chicago Sun-Times]
When my son’s apartment was burglarized in March and his MacBook Pro stolen, I immediately wished it had some kind of tracking program installed on it. [Houston Chronicle]
Dallas Police are looking for two men who were caught on video burglarizing a music store and later pawning the stolen instruments. [Dallas Morning News]
And British publications use burgle with no humor intended:
They are also charged with conspiracy to burgle Bush Barn Farm and conspiracy to pervert the course of justice. [BBC]
A bride-to-be who burgled her friend’s home has had her curfew lifted so she can stay out late on her wedding day. [Daily Mail]
In Switzer-land, where users participate in a drugs programme, there has been a 90% fall in addicts burgling homes. [Mirror]