Enharmonics are just the notes with two possible names, i.e., C# and Db. Which one you use will depend on the key of the music. For example, in the key of A major – C# will be the third. In the key of Ab major – Db will be the 4th. You could call it whatever you like but for music reading and for communication among musicians it’s best to try sticking to common convention.
Double sharps and flats come about mostly because of music notation. In any key of music we try not to repeat note names. For example, In the key of D we have the notes D E F# G A B C#.
In the key of D# we have D# E# F## G# A# B# C##.
If we wanted we could call the F## a G and the C## a D. The problem here is when this is written in notation we’d have to put the F## and the G# on the same stave line. This can get messy because we’d have to use a lot of accidentals … much cleaner and easier to read if we use the double sharps (or double flats).