In most contexts, they are opposites. So you could say: "I will DO nothing else", or "I will NOT DO anything else".
In your question, it's not clear if you are the customer or the shop assistant.
If you are the customer, you could either say:
"I will pay this amount. Nothing else." or, more commonly "I will pay this amount. No more."
If you are the shop assistant, then the first part of your sentence would need to change. For example:
"The total cost is this amount. Anything else?" or
"The total cost is this amount. Nothing else?"
So, if you are using the two words on their own as a question, they are both valid.
"Anything else?" = "Nothing else?"