Skill breakthroughs are obtainable by using skillbooks (obtainable from events or by using ninja fragments). Skill trials are obtainable from using experimental vials on a ninja (skills you don't use protection powder on will be rerolled). You can switch between corresponding skill trials and breakthroughs whenever you want.

So, if a ninja has both skill breakthrough and skill trial available, it's better to use some experimental vials before buying skillbooks with ninja fragments to finish said ninja's breakthrough/trial.