In most cases, 'begin' and 'start' are synonyms, and can be used interchangeably. For example, 'When does the lesson start?' and 'When does the lesson begin?' have exactly the same meaning. However, you can only use 'start' in the sense of making something start working. For example, you can say 'He started the engine', but you can't use 'begin' in this sentence. You can also say 'He started a business', in the sense of establish a business, but you can't use 'begin' here either. Nor can you use replace 'start' with 'begin' in phrasal verbs such as 'start off' or 'start out' and so on.

The two words originally had two different meanings going back in English history, but now-a-days are used to mean the same thing. You will hear start used a lot more than begin in everyday speech.