Garen is my main, but I also play a lot of Irelia, that said, Garen is rather easy to counter (kite and armor). Irelia is much more potent, as said before, gap closer, stun, and free heals and true damage
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I used to main Irelia before the nerfs, after which she was almost unplayable against the most common tops, as they could easily outharass and outsustain her, although you could occasionally outplay people still.
With the recent buffs, she's pretty stong early, but drops a teensy bit around level 4 or 5 until 7, so be careful there. At lvl 4 where you get your stun, if your jungler has good timing and you're playing Irelia right, you should be pushed to your tower and set up for an easy forced flash, at the very least. If you get a kill, you should be able to snowball hard from that point, and if you don't, try to farm until you hit 7, where you should have enough for Wit's End, be able to force your lane to back or get a kill, and start helping your team power through midgame.
Irelia beats Riven, Nasus, and Garen early with armor pots if managed well. Watch creep waves, force them to push the lane to sustain/farm, farm under tower, and wait for mistakes or your jungler if things go sour. Against Garen, take advantage of his long cooldowns. Against Nasus, you can do whatever you want until he withers, then dash away to a creep. Against Riven, watch for her dash, and back away if you see her go to use it. She'll likely be using it to try to harass you going in to cs, so back away and q to a creep when the shield's gone. If she decides to fight then, you can outharass her. That shield along with that one extra empowered auto lets her win trades with you. That's it.
Remember not to use q to get to champions when you want to harass. Wait for them, or a time they're close, then activate hiten, auto, q, auto, e auto. Try to do this when your creep wave is larger than theirs. Doing this with a competent jungler ganking is an almost garunteed kill with ignite.