Bleaching is to get white/light blond/blond colour. Dying can be of any colour: Brown/red/black/pink etc.

Bleaching is more damaging. But depending on what colour you desire you'll sometimes HAVE to bleach. Some colours aren't caught by the hair, mostly exotic colours like blue, pink, purple etc. and for some people any colour lighter than their own natural hair colour is difficult to have.
To get these colours you need to both bleach and dye. you bleach then dye.
It is like bleaching paper in order to colour on it.
Dying your hair only few degrees lighter/darker than your own hair colour is over all less damaging.

I'm a brunette. Not Asian Black, But not Very light either. Most people can't get red dye caught by the hair without bleaching it first. But I never bleached my hair, and been dying my hair with all degrees of red for the past few years. It works for me.
So I suppose it isn't only about the natural colour. But also type of hair. Ask your hair dresser for advice.

On the other hand DIY is much much much cheaper. and way too easy if you're getting your whole hair dyed with the same colour (no high lights). You mix the gradients in the dye box like instructed and apply it to your hair just like shampoo. LOL. I have been dying my hair by myself for a couple of years now.
Bleaching better be done in a saloon though.