The clear cut answer is that each is better in different situations.
Fighters do anti-light damage. This means the get a bonus against bombers, colony frigates, long range frigates, and scouts. They are ok against other fighters.
Fighters, however, suck against anything with medium armor and up. In the upcoming patch, they're going to do 50% instead of 25% against medium and up, but still, it's godawful.
Bombers do anti-very heavy damage. They are good against buildings and heavy cruisers. It's worth nothing that they're fairly strong against capital ships too. They're not too bad against heavy and medium armor.
Bombers, however, are very vulnerable to fighters. They can actually take a good bit of punishment when up against flak. They still die, make no mistake, but they'll last way longer against flak than fighters will.
Which one is best? I'd have to say it depends on what stage of the game you're in. In early games, people make scouts, lrf's, and colony frigates, so fighters are best early.
Once you go on the offensive (since you'll be up against enemy tactical structures), bombers are best. When hc's pop up, then bombers are best again.
Still, what happens is late game you should at most be doing 1 fighter squadron for every 1 bomber squadron, for the sake of balance. Early game, just get fighters, but do pick the bombers if a planet has the heavy cruiser pirates.