Unsatisfied means that your state of satification has not been reached. He ate a big lunch but his hunger was unsatisfied. There was nothing wrong with the food, there just wasn't enough of it. Had there been more food, he would have been satisfied. Dissatisfied means that what was offered did not please him at all. He was dissatisfied with the food means that it was poor quality, it was too expensive, it fell short of the standard he was expecting. It is worth understanding the difference in meaning between 'dis-' and 'un-'. I don't know how many times I have seen 'uninterested' and 'disinterested' misused. Good luck.

this might help using a different word. if you pass an exam you became qualified to do something. if you misbehave or commit fraud or break the law you became disqualified. Qualification taken away (DIS) from you disqualified. if you fail the exam to become qualified then you are unqualified (not qualified) HOWEVER there are words that begin with dis that can also mean NOT in the sense of UN+(word). disarray, not organised or chaotic. Disassemble take apart can also be considered as not assembled after the disassembling. Disorganised = not organised. and other words. see the second example negative DIS then scroll down and see a list of examples https://www.englishhints.com/prefix-list.html

the subtle difference for these two affixes are 1)un+(word) = not 2)dis+(word)=apart away from unsatisfied = not satisfied. disqualified= colloquially people use it as not qualified. But it means your qualification was taken AWAY FROM YOU. that's why you get DISQUALIFIED in a driving court and have your licence to drive a car/vehicle taken away from you. EDIT: i will attempt to try an example. you can be unsatisfied for having bad customer service. Forcing you to leave a restaurant feeling unsatisfied. without having attempted to eat or order a meal. (not satisfied) or you can report being dissatisfied to the management. For not receiving the expected level of customer service. (feeling forced away apart from feeling satisfied). There are also words beginning with dis that also have the meaning of not. see part two below

I tend to use unsatisfied when describing something like food service at a restaurant perhaps in this manner of "I am unsatisfied" or "he/she is unsatisfied". However although you did not ask about the phrase "not satisfied" I find myself using that term when at a place of business or talking on the phone about a billing account or service I am subscribed to. I would say that "dissatisfied" comes out of my mouth the least. I cannot think of a reason to not use dissatisfied. It is fluent English to use either one of the two.

I can be satisfied or dissatisfied with something: this describes how I feel about whether or not it met my expectations. It describes a mental state. If am old enough, I can vote; if not, I cant, so if I am 59, the age condition is satisfied, if I am 15 it is unsatisfied. The satisfaction here is an objective statement of fact, with no mental component. A requirement can be satisfied or unsatisfied, but it cannot feel satisfaction, so it cannot be dissatisfied.