SQL - I have just Differential backup but database exists
You are pretty much out of luck unless you have a full backup previous to the differential backup you want to restore. Your backup chain is incomplete if you do not and with that your data is simply gone.

It sounds as if you received the error for the need of a full backup, and then took a full backup and restored that, then your differential. Sorry to tell you that is not going to work either.

Your only hope is that the data can be recovered manually because you are not going to be able to restore that database with out a previous full backup.

I would highly suggest you adjust your backup process now so this can be prevented. You need to ensure your backup chain is complete in order to provide a proper recovery process (along with ensuring integrity).