I used to do karate and have recently started boxing. Now obviously the techniques all look very different and the stances etc. However I wonder if karate punches may have more power to them or stuff like that.
This may be an unsatisfactory answer, as I am no expert in boxing or karate, but I have experienced multiple arts and sports and have noticed something:
They may have minor differences, or they may not have any differences at all.
This is because there is a thing called body mechanics. Basically, there are only so many ways you can deliver a good punch, kick, or other move. It is very possible that multiple people have independently discovered the optimal way to punch. I would therefore advise you to keep your eyes open and see what karate and boxing share. They do have slightly different goals, so that will affect how the techniques are delivered.
On a related (and mildly humorous note), I have found such a connection between HEMA (Historic European Martial Arts) and baseball. There is a swordfighting stance called posta di donna. You can get a lot of power out from this stance when swinging. Batters use pretty much the same stance when they go up to bat. It's a biomechanically similar setup, just used in different contexts and executed slightly differently. If Fiore dei Liberi saw a baseball batter, he may ask "why is that man in woman's guard? Who is he going to hit?"