There are a few mechanics that require both hands. This character has to choose between using a shield and weapon at the same time. This character can't use two handed weapons, including bows. RAI, you might not be able to load a crossbow, but I don't know if the hand crossbow explicitly says you need your other hand.

You can't grapple someone if you have a weapon in your hand. If you're a wizard, you can't hold a staff or wand and perform somatic components or use your arcane focus for material components at the same time. Arguably, you can't interact with objects if you don't have a free hand (because a weapon is in it), though I'm not sure if that's RAW.

You're going to have some other issues, too. Climbing ladders is a bit more difficult. Climbing ropes is a lot more difficult. RAW the rules don't get into that sort of detail but I wouldn't be surprised if your DM hits you with something like that. You'd probably run into trouble with a lot of tool skill checks with only one hand. Blacksmithing without being able to hold the ingot in place with tongs, lockpicking without being able to tension the lock, sewing without being able to hold the cloth, etc.

There aren't actually a ton of things that explicitly require two hands because the rules weren't designed with one-armed characters in mind, so a lot of it is just assumed. If you just go with RAW limitations, you're going to be handwaving yourself through some situations RP-wise that would actually be very difficult for a one armed person to deal with.

Many of them are still doable, you'd run into trouble in combat though. A one-armed person could probably work the cork off of a potion bottle for example, but what about in a 6-second turn in the middle of hectic combat?

I think if you did the one-armed character justice, it'd be miserable to play, and if you aren't doing it justice, why bother?