Full scholarship and student advisor relationship
The main effect of external support on a student/advisor relationship is that there is less external pressure for production of short-term results.  Whether this is a good thing or a bad thing depends strongly on the individual student and advisor.  


With less pressure on the student, there is more freedom to develop a unique research agenda but also more opportunity to get "lost" in the possibilities and waste time.
With less pressure on the advisor, there are fewer boss/employee dynamics in the relationship and generally less strain, but also more opportunity for the advisor to be distracted by other short-term concerns.


A secondary effect is that the student may have less opportunities to be apprenticed in grant-writing, since their support is already assured.

In sum: with a good advisor and a student with a mature outlook, the freedom of external support is a blessing.  If either of those two ingredients is lacking, it can be a problem to lack the forcing functions of external support.