I don't think we've ever had a head-to-head comparison of partial profile (PP) and ACBC, so the response below is a bit more theory-based than I would like.
First, PP will give you estimates of utilities of all levels of all attributes. Although the PP design is less efficient (usually about 50% as efficient as full profile) it also contains less response error (on average it's about 4 times as efficient as full profile in terms of response error) so that in total PP is about twice as efficient as full profile. PP (like ACA before it) tends to make your utilities less spiky (i.e. larger utilities won't be AS MUCH larger as you would have in full profile and smaller utilities aren't AS small as you see them in full profile designs). This has particular implications if one of your attributes is price, because price tends to be highly influential and with lower utilities it will look less important in PP than in full profile (and it will imply larger willingness-to-pay estimates). Of course if price is not one of your attributes this last point will be less relevant to you.
With ACBC I would probably ask a question prior to the ACBC questions where a respondent chooses 10 (or up to 10) attributes that are most important to her. So Smith might pick one set of 10 important attributes and Jones might pick a different set of 10. You build a constructed list of the items chosen in this prior question and it becomes your list of attributes for ACBC. ACBC will give your unincluded attributes will get utilities of zero unless you specify that they are missing at random. Though they are NOT missing at random, since they are specifically disincluded for being less important, it’s also likely that they don’t have utilities exactly equal to zero, either, so you may want to do some post hoc adjustments to put them somewhere in-between.