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: 
With the guitar being a positional instrument, meaning one can play the same tune in many different keys but retain the same fingering and strings, merely moving where on the neck the tune is played, then knowing note names as the tune unfolds is not necessary. The relationship between the tonic and other notes, as far as where they are relatively speaking, fret and string, is easier to remember. In fact, there is often no need to reference from the tonic, but from the last note played. A lot of the time the next note will either be on the same string or an adjacent one, so knowing the various scale patterns is far more useful for improv. than the names of the notes. That's the academic side which (in my opinion) doesn't often come into play (sorry for the pun) during improvising.
I'd memorize the relation of the roots to the tonic for analysis and overview. However, I find it more musical, if I have to try and verbalize, to remember the relation of a note and the current (and sometimes previous and/or next) root.