I couldn't know the difference between the two verbs and I couldn't understand also what does mean "Kosten abrechnen" how can costs be settled?
Abrechnen means to settle or clear in an accounting or financial context.
The mentioned "Kosten abrechnen" could be used when you are a contractor and finished a project for a customer. Now you present this customer with your costs and ask him to pay them in addition to your fee. The process is also called "Kostenabrechnung". The settling here is that you forward the costs to the customer, and thus they are not your costs anymore. Your expenses are equalized after that.
Ich werde mit dem Kunden die Kosten abrechnen.
It can also be used figuratively. For example if someone has been treated badly and then takes revenge either physically or verbally.
Stefan hatte angekündigt mit seiner alten Firma abzurechnen. Und tatsächlich, zwei Tage nach seiner Entlassung rechnete er in seinem Blog mit seiner alten Firma ab. Auf zehn Seiten beschrieb er detailliert wie seine Firma ihre Kunden betrog. Was für eine Abrechnung!
Just the normal verb for calculating. It has additional meanings in other contexts of course.