If it's obvious that it's ME who'll see you next time then I just write, See you next time. Four words versus six (I'll = two words) Saves ink/space and effort!

The first one is technically not a full sentence, and is just a short form of the second one. As you know, English (as opposed to Japanese) grammar requires an explicit subject for all clauses. That said, there are a limited number of set phrases where we can drop the subject in *informal* speech.

Adding in “I’ll” makes the sentence slightly more formal. It’s sort of similar to using じゃあ(ね？) Instead of さようなら Or maybe a more correct comparison is: また後で Instead of 後で会いましょう