I play both because for the ten missions the KDF get to themselves are mostly entertaining. But that passes quickly and I'm stuck with a lvl 50 warrior with little to do but the contraband farming he's so good at.
My Feds have a lot to do, and I can set specific leveling tasks for them. Eng for Foundry, Tac for patrols, and Sci exploration. I've done the mission content enough now that I only use them to level up when I'm sick of seeing the same tiny ship all the time.
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I have multiple characters for both factions. But my mains are Fed side. I play the federation because to me the federation is star trek, all shows are always from the federations perspective and i just like the federation style of ships. And i always thought of the Klingon's as emo crybabies who always whine about honor and never really show any traits of honor, that and cowards who hide behind their cloaks.
As for the actual KDF faction in game, i do like their story missions, they are rather interestingly done for the most part and KDF maps generally have a good visual look, but the lack of customization options and that absolutely retina destroying red UI are major turn off's.