[verb ます-stem] + 出す
出す after the ます-stem of a verb says the action starts, usually suddenly and without warning: なく泣く becomes なきだす泣き出す, "burst into tears". The compound conjugates like 出す on its own: はしりだした走り出した, はしりださない走り出さない, はしりだして走り出して.
Both mean "start doing". 出す puts the weight on the moment the action breaks out; はじめる始める simply says it began, and suits a planned or gradual start.
出す reports a start that happens or has happened. It does not go into a request, an invitation or a statement of what you intend to do; 始める does.
These go beyond "start suddenly".
In some compounds 出す keeps its own meaning, "out": とりだす取り出す is take out, とびだす飛び出す jump out, にげだす逃げ出す run away. These are separate words, and a dictionary lists them.