Molt and moult are different spellings of the same word, meaning (1) to periodically shed an outer covering and (2) the act of shedding an outer covering. Molt is the preferred spelling in American English, and moult is preferred outside North America. Both spellings are used in Canadian English.
Both spellings are old, first appearing around the 16th century, but moult prevailed by a large margin until the early 20th century, when the shorter spelling surpassed the longer one in American English.
[T]here is so much variation in the appearance of any species—depending on gender, age and stage of molt—that no one guide can get it all. [Wall Street Journal]
When the caterpillar molts, it stays put for a while instead of traveling up the tree to eat. [Los Angeles Times]
In recent weeks, the Canada geese have begun their annual molting, meaning they could not fly. [New York Times]
Outside the U.S.
[C]entral heating causes them to moult all year round, causing their fur to clog up their digestive system. [Telegraph]
Its need for calcium to build its shell through the many moults of a lifetime makes the limestone midlands the crayfish’s chief territory. [Irish Times]
This is usually done when the birds are moulting and cannot fly, so they can be rounded up and killed. [New Zealand Herald]