It simply a set phrase. New Oxford American Dictionary the —— the better used to emphasize the importance or desirability of the quality or thing specified: the sooner we're off the better | the more people there the better. be better off be in a better position, especially in financial terms: the promotion would make her about $750 a year better off | (as plural noun the better off) : a paper read mainly by the better off. the better part of almost all of; most of: it is the better part of a mile. better safe than sorry proverb it's wiser to be cautious than to be hasty or rash and so do something you may later regret. better than North American more than: he'd lived there for better than twenty years.

If someone is 'better' it implies a change in themselves; like they have become a better person. If they are 'better off' it is a change in their circumstances that benefits them.

“Better off” is often used for money, but can be used for other things. “I was planning to take the train from the airport” “Don’t do that. To get to my house, you’ll be better off taking a cab.” “She was trouble. He’s better off without her.” “He’s better off financially in his new job, but I don’t think he’s as happy. His hours are too long.” Which trail to the summit do you like? Today, you’ll be better off taking the one through the forest. It’s going to be hot and you’ll appreciate the shade.