They basically mean the same thing but there are nuances. As a result, there are times when you would use them in different contexts. For example, you would 'start an engine' but you wouldn't 'begin an engine'. 'Start' goes with 'finish' like in races and board games e.g., 'start on go' rather than 'begin on go' "Begin' goes with 'end' like when reading a book e.g., typically one would say 'the beginning and ending of a book' and not the 'starting and ending of a book'. There are many more examples.

When used as verbs, both are usually interchangeable. However, "start" can be used as a noun as is: "the start of the Olympics". "Start" can also be used to imply causation, such as: "I start a fire". "Let's begin" and "let's start" are mostly identical phrases, but "let's start" sounds slightly more casual.