Well, actually, there is a difference (albeit slight). "Burst into tears" is usually a sudden outburst. One minute, he's fine and the next minute, he's gushing tears. That's why we use the word "burst". "Broke down in tears" doesn't necessarily mean that sudden burst. He may have started with a little trickle and it may have slowly increased and suddenly, he can't speak because he's crying so hard. He did not "burst" into tears but wound up "breaking down" anyway.