Coliseum and colosseum are both common spellings of the word referring to (1) the famous Roman amphitheater built in the first century A.D., and (2) any large amphitheater used for sports or other public events. Neither spelling is considered wrong in either use, but while the forms are often used interchangeably, the famous structure in Rome is now usually spelled Colosseum, and coliseum is generally reserved for other uses. Exceptions are easily found, however, and there is no consensus evident in popular usage.
Some English reference sources claim that colosseum is the British spelling for both the Roman structure and the common noun, but this is not consistently borne out in writing searchable on the web. In the rare instances where the word is not part of a name, coliseum is about twice as common as colosseum on British websites. But because the sample size for this is small, it might not be a reliable indicator of where British usage actually stands right now. It could be that colosseum is indeed the preferred form in all types of British writing that are not searchable online.
The issue is clear-cut in the U.S. Coliseum is the preferred spelling except when it comes to the Roman Colosseum. As in British writing, the word usually appears in names of buildings, but the generic noun is a little more common in the U.S. than in the U.K., probably due to a spate of new buildings with coliseum in their names built over the last few decades.