What words distinguish them? ..
In British English a term for someone who takes away your garbage to dispose of it is 'dustman'
A man employed to remove household refuse from dustbins.
Also British English, the rather lovely, and sadly now very much in decline term for someone who takes away your rubbish to try and sell it is 'the rag and bone man'
An itinerant dealer in old clothes, furniture, and second-hand items.
There is a US tradition of the rag and bone man, a junk collector. They don't pay for what they collect, but rather sell it along to recyclers. In my distant youth I would see them driving pony-carts.
You may also hear sheeny man in some locations but this has an unsavory history as an anti-semitic slur (though I doubt people using it today would be aware of that).
One term is gaining popularity in my region (central USA) due to the History channel show "American Pickers" - The person would be called a Picker.
a person or machine that gathers or collects something.
In this case, they are gathering "junk", sometimes to collect, but also sometimes to gather and then sell.
There may be a slight difference however, in that a picker is choosing, or picking, what to take from you - only the stuff that he wants or is able to sell.
What words distinguish them?
None in my locale.
we have "bin-men" who empty our non-recyclable bin into what we used to call a dustbin-lorry. We also have "bin-men" who empty our recycling bins into a separate vehicle. The local council arrange for both collections.
I suspect you'd have to talk about bin-men a lot before the need for additional qualifying adjectives started to seem tedious enough that you'd feel a strong need to find or invent a new noun.