They both mean the same thing. I feel you can use them in different instances. No problem is a quick response to thank you. Not a problem can communicate that a task or request was easily done without any issues.

‘No problem’ and ‘not a problem’ in this context both mean the same thing basically that everything is okay.

‘No’ and ‘not’ are both negative words used to give the word ‘problem’ the opposite meaning. However ‘No’ is being used as an adjective to describe problem and ‘not’ is being used as an adverb to describe ‘a’, an article which is also an adjective as it describes the word ‘problem’

So It’s important to note the importance of the form of these phrases. You cannot say No *a* problem or Not ** problem.

‘I appreciate the input’ means I am thankful for your ideas.

It sounds as if in your email you expressed your opinion about something and also some doubt as to whether it was appropriate for you to do so. In that case “Not a problem” would be short for “It was not a problem that you contacted me. “Appreciate your input” means that he was happy to hear your viewpoint, but he isn’t saying that he agrees with you or committing to doing things your way. “I know you are considering two candidates for the open position. I’m not sure if it’s appropriate for me to do so but I wanted you to know that I felt the second candidate would be perfect for the job. He got along very well with everyone.” “Not a problem.” (Sounds very slightly better and more thoughtful than “no problem”) “I appreciate your input” (I’m glad that you gave me your advice. I will take it into account when making the decision)

"not a problem" and "no problem" have essentially the same meaning and could be used interchangeably. "Appreciate the input" would mean the person is thanking you for information or advice you provided to them to assist them in a certain circumstance.

"not a problem" and "no problem" have essentially the same meaning and could be used interchangeably. "Appreciate the input" would mean the person is thanking you for information or advice you provided to them to assist them in a certain circumstance.