I'm not aware of research comparing the two head-to-head. As co-inventor of partial profile conjoint I am probably a bit biased but here is a comparison of the two approaches:
PP is a much faster interview for respondents, it dramatically reduces response error compared to CBC and it can handle many more attributes without overwhelming respondents. It indirectly encourages deeper processing of preferences because a given attribute (including those that are most important) only appears in a minority of choice sets, allowing other attributes a chance to exert influence.
ACBC is probably better for handling price than is PP and it directly encourages respondents to process preferences more deeply by trying to detect, and then ask respondents about, potential non-compensatory effects. ACBC may also prevent some of the "flattening out" of utilities that we sometimes see with PP models.
I hope this helps.