What is bitcoin mixing/laundering for? What is the difference if I'm using bitcoins that were given me by somebody I sold product to, or by some mixing bitcoin service?

After all, aren't I using those bitcoin at the same place and ordering what I want with the same name and address?
Bitcoin transactions are traceable, public and are recorded permanently. Although, there are some ways to minimize that, by not reusing addresses and have multiple wallets for different purposes. But even that is not foolproof because there are analysis services that can group addresses through data crunching.

Now imagine you get coins from someone that got them from someone, etc. Nothing wrong with that right? Unless, those coins are part of a bunch that were hacked from an exchange or traced back to a payment for some illegal activity.  And some exchanges start blacklisting them.

A mixing service tries to remove that linkability/traceability.

IMO you would be better off just using a crypto currency with unlinkability and untraceability built into the protocol in way of a mixer.