I think if you cut, people see it as a "Oh, just another cutter" kind of thing and kind of gloss over it. It's weird to think that it can somehow have become some sort of weird-not-quite-norm thing. But if you burn and people see them or find it, they have more of a fascination with it because typically, it can look worse.
In my experience, I think burns get a bit greener looking than cuts and are more easily infected. It kind of depends what you do them with as well though. I used to do quite a lot with different things - cigarettes, curling irons and I only did it for that immediate singe of pain and the hotness of my skin compared to how I felt so numb otherwise. After I'd got my 'kick' out of it and my mind had settled down, I always regretted it because it hurt twelve times more than it did at the beginning and the marks were always there longer. After saying, "Oh, I caught it on the oven" or "Flipping hair curlers" a few times if they got spotted, people got suspicious 'cause cuts are a bit more 'everyday injury' than burns. I mean, you probably accidently burn yourself once a year on the oven door or something, yeah? But you walk past something and catch your arm on a tree or fence or scrape it on a wall or all that kind of stuff frequently, right? So, mhmm, I guess people did react differently to them. But the burns always seemed to 'repulse' people more than anything else and it just made me feel even worse.
I can't really help with taking care of them because I never did. I just left it alone. Maybe put a bit of antiseptic cream on if it looked a little too red around the edges but other than that, nothing.