Changing the game speed changes how many turns it takes to build improvements, build buildings & units, and research techs. My math may not be perfect, but if a tech takes 3 turns at Quick to research, it'll take 6 turns at Normal, 12 to research at Epic and, I believe, 18 at Marathon (never play Marathon).
The slower speeds have more turns in a game to make up for it. Very few people play Quick. Marathon isn't scaled quite the same - units are cheaper relative to buildings at Marathon.
Where the differences comes in is that units move the same # of tiles, regardless of game speeds. Therefore, on Marathon, you may have everything explored by the time you research your third tech.
This has 2 consequences for war. After you build a military force, it takes a longer time to go obsolete at slower speeds. At quick, you can build an army of Swordsmen and, by the time they get to the enemy, Macemen are available. Generally, the slower the speed, the better for a warmonger strategy. Also, faster units are more valuable at faster speeds. This definitely includes India's Fast Worker - the best UU at Quick, practically worthless at Marathon.