Not only does it depend on the snake (I'm sure you wouldn't want to be bit my a mamba!), but if you're allergic to anything in the saliva. For example, watersnakes have a mild anti-coagulant in their saliva which causes a bit more bleeding than the average bite. If the bitten person has a clotting disorder, this can cause serious issues. I was bitten by a foot long watersnake a few years back. That bite bleed more than any of the bites from my adult corn snake and I'm healthy without any clotting disorders.

Another example is rear fanged snakes. Most of the ones that's common in the snake trade won't cause any harm. Maybe some minor irritation at the bite site. However, if the person bitten is allergic to the venom, then there's cause for alarm.

Yes, with snakes suck as corns, boas, pythons, etc. It's nothing to worry about. Maybe some bruising or superficial puncture marks if it's a larger snake, but there are species to be wary of. To say that people shouldn't care about snake bites in general is at best, foolish.