Solved is a status that you as an agent give a Ticket when you think that the Ticket is ready to be closed but could have further questions from the requester – a Ticket that is 'on hold', so to say. If the requester would answer the Ticket again, the Ticket would be reactivated with the status Reply.

You can also set status Solved on a case that requires further action from another party. Let's say that you work in an online store and that you get a Ticket where the sender wonders when he or she gets his/her money back, after returning an item. You can see in your system that the product has been returned and that this should have been stated by your partner for payments over two weeks ago – yet apparently the sender has not received his/her money yet. You then choose to reply to the sender that you apologize, and you forward the case to the other party. When this is done, you set the status Solved and leave the Ticket until you get a reply from the other party – you can not do more in this case at the moment, but need the other party to look further at this and answer you or the sender. So, the Ticket is not completely resolved but requires further action, at the same time as the case is solved with you.

Giving a Ticket status Solved, is useful when you wish to have the history if the requster answers you again. 

Closed is a status that you give a Ticket that you know for sure that you can close, manually by you as an agent or automatically after a certain amount of days (that you've chosen). Future e-mails from the same sender will then land in the client as new Tickets, without any history. (You can still change status on a Ticket that has the status Closed.)

Read about all the differents statuses in this article: What do the different Ticket statuses state?