What use do the spearmen have in Settlers 3?

As a rule of thumb, never build swordsman. Spearman might be slower than swordsman and therefor are less effective vs archers but they are just for tanking damage and the extra damage they do when they are in range is much more effective than the speed of swordsman. In the end archers win battles, at least 3/4 of your army should be archers, the rest spearman as your army grows the amount of archers should grow/increase relatively to spearman.

Source: I have played settlers 3 online for a long time, I actually stood in the top 10 ranking for a long time. I played mostly big and long random games with peacetime of 45+ minutes though.

I'm sorry but you are wrong on your statement there.

Spearmen are stronger than sword fighters, if your barracks buildings are as evolved as your enemies you will notice that archers are strong against spearmen (spearmen are slow), spearmen beat the hell out of sword fighters (their melee is stronger) and sword fighters beat archers (speed wise, they can get into close contact very easily).