Begin and start are synonyms - they have the same meaning.
'Begin to' and 'start to' are also synonyms.
That is what sara was saying when she used equals (=) - they are equal in meaning (she could have been a little clearer).
Start has additional, less common meanings which are different than begin. For example,
to 'start' can mean to 'move suddenly' - "He started when he heard the thunder" - means he jerked/convulsed/moved suddenly.
But for the most common meaning, begin and start mean the same thing.