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.

So the actual breakthrough is the real thing while the trial is just testing out the skills?

They are both real. Skill trial style is using vials to experiment, like the purify equipment, the result is completely random. Normally people use skill trial to get as many +1, +2 or Y spells, then use the book to upgrade further.

Much like the purify equipment, getting a random +2 or Y is like getting a purple or gold (lower chance). So when you do get +2 or Y or Y+1, people will use protection to lock it.

Fragment for expensive ninjas are much more expensive so when available people use skill trial to attempt to "break" the skill.

I understand now... preciate it