They are very much interchangeable ,the 2 words,( to begin with, to start with),when in the sense of initiating an action ,however sometimes it is more appropriate using the one over the other. When having other senses you can't replace the one with the other. Moreover when associating one of those verbs with a preposition in the form of a phrasal verb ,they can't be interchanged,also when they are used in certain expression modes. Examples: - to start up the computer (phrasal verb : start up) you can't say : to begin up the computer. - As in intransitive verb 'to start' someone doing something means to cause someone to do something : What he said started me thinking. You can't use here 'began' me thinking. - In the sense of switching on a machine. "He started the engine. ( not began the engine) - 'to start on something' means to begin in working or dealing with something. "She started on the cakes" ( started eating them). Not interchangeable with 'begin..' - 'to bring into being' meaning to arise or originate ,you can't use 'to start into being'

Start to me is alot less formal. Examples : '' Start on such a task. '' '' Begin the examiniation. '' I'd say Start would be the more appropriate one to use in 95% of situations! Sorry for the vague answer. :)