Question: When playing guitar,especially improvising,is it best to think of the notes a actual lettered notes or Scale degrees related to a tonic ? Since the keys are easily movable , I'm thinking that its better just to know the degree you want to play and knowing where that is related to the tonic.

Answer: 
Every guitar player is different so it depends. I have a friend who took piano lessons and read music before learning guitar and she would much rather know the names of the notes. 

I play based on sound and feel and knowing where to put my fingers relative to where they are now to get to the note I want next relative to the note I am on. 

But I have a musical ear. If you want me to play something, I would rather hear it than see it on tab. If you give me standard notation and ask me to play what is written I will get back to you in a few days. If I hear it, I can play it back immediately. 

I personally have no need to know the names of the notes I am playing while I am playing them (I **could** tell you if I **thought** about it). I just need to know how to make what I hear in my brain translate to my instrument and the easiest way for me to do that is to start on the right note and go from there. 

I am quite certain that if I was improvising and I tried to **engage the part of my brain that would allow me to identify the notes** I was playing, it would stifle my creativity and instinct. I think the best improvisers play by **allowing their mind to engage with their soul and play what "feels" right. Thinking about the NAME of the note they wanted to play next - would short circuit that connection.** Who cares what the note is called as long as it sounds good!

If you can improvise without knowing the **names** of the notes you are playing, your time will be better spent improving your craft by playing more - than by **trying to learn to identify each note you are playing while improvising.** That is my assessment based on my own personal playing style. Your mileage may vary.
I think it would be better to think the scale degrees. 

For instance, let's say you play E (V) and then Am (i); if you think in scale degrees, then you will lead the notes that consist the V (E major chord) to the tonic chord (A minor). This way, you make the listener believe that there is some connection between the notes you play. They are two degrees of a scale.

If you only think the notes, then it might not give the best sound outcome (the chords might not sound connected with each other). 

The notes will be the same, no matter how you think them, but the way you think, does affect the sound outcome.