STAR Voting is designed to maximize both utilitarianism and finding majority supported winners where possible. I look at it like a debate between quality and quantity. Both are important. In STAR Voting the scoring round measures quality of support, (how much do the voters like the various candidates. Then, the runoff measures quantity or number of supporters, (between the two front-runners, which do you prefer.)
As for why I believe STAR Voting is more fair and representative compared to Score, of course I have to start with the disclaimer that Score is a very good system, and they get the same winner most of the time, but in Score Voting if I vote honestly and don't give any front-runners a top score, then my vote's impact is less than if I had strategically given my lesser-evil a top score.
Another shortcoming with Score that is addressed by STAR is if some voters fail to use the full scale. Strategically speaking, the best strategy in both methods is always to give your favorite 5 stars, but it's to be expected that some voters will give a mediocre score to a mediocre favorite, especially if they are new to the system. Unless you normalize scores, Score voting gives voters a chance at an equally weighted vote, but doesn't actually guarantee it. Hopefully this will be corrected with voter education and good instructions, but with STAR there's the added failsafe that the runoff is binary. Ultimately your vote is just as powerful as everyone else's.
These voters, voters who are currently marginalized in our current system, should have just as powerful a vote as a voter who does support a frontrunner. Guaranteed. STAR Voting does that. If your favorite can't win, you can give your favorite 5 stars, give your lesser evil 1 star, and that will still ensure that if it comes down to it, your fully weighted vote will help prevent your worst case scenario. That's how STAR Voting prevents tyranny of the Majority. If your lesser-evil is actually substantially better than your worst case scenario you can give them a better score.
As far as I know STAR may be the only method where even if none of the candidates you like can win, your vote can still make a difference and help prevent your worst case scenario.