They are basically the same thing. If I was being pedantic I'd say that to feel the need is more of a compulsion. You 'have to' text all the time, or you can't stop doing it. Whereas to feel like texting all the time is a bit softer. You would like to do it, but it doesn't control you.
I feel the need to breath all the time.
I feel like eating cheese all the time.
That is semantics though and they would both be read in the same way: I can't stop myself wanting to text people constantly.