Being "better off" means that things (other than yourself) are better for you. That is, you are able to be happier, or be in a better condition. For instance, being "better off" without Steve means that without him, your life would be better. Notice the distinction: Your life is better, not you yourself.
Being "better" means being superior in some way. That is, being smarter, stronger, kinder, etc. Merely being happier does not count as being "better", unless that happiness is useful for something. e.g. Seeing you happy makes others happy, therefore your happiness makes you better for them.
If you're "better" without Steve, it would imply that you yourself have improved somehow. Like, perhaps being around Steve made you a less patient person, or made you physically ill. And now, with Steve gone, you are better to others, or are capable of doing more than you could before.
The key here is that "better" does not mean "happier". It means having a functional superiority in some aspect. That is, it is a change within the person (or object) which is "better". Benefits from being "better" could easily apply to people around said person, rather than the person who is "better".
When adding "off" the meaning changes to having a functional benefit for the person who is "better off". The person is unchanged, however (Save, perhaps, for being made happier).