Certain Civs are Wimpy?

I am just trying to play as Dido. I hit a barb archer with a city attack and then a spearman and it was still alive. I know I've played other civs where my spearmen would just walk through archers with one hit.

I know... I might have had some bonuses or something, but it just seems like certain divs troops are just weaker than others. My imagination?

definitely imagination it is the same with all civs.

All civs have exactly the same units barring UUs, so this is all in your imagination

The Oligarchy SP, as well as the Honor boost against barbs, make a big difference in how quickly you can kill barbs. That, + your units' upgrades probably accounts for all the difference you're seeing.

Pay attention to the terrain you're fighting on, going to be fighting on, and what's around you. If you're fighting an up hill battle, you'll do less damage then if you were the one I the hill, or even both on hills. Forrest's and jungles provide more cover for all targets, more hits.

Pay attention to your upgrades. Rough promotion melee, attack and occupy rough tiles. Rough promoted archers, shoot people occupying rough tiles. Vice versa for your open terrain promoted troops.

did you change difficulty between games? One of the things difficulty governs is how much damage you do to barbarians, and how much you receive in return