Hi Daniel. Yes, the meanings overlap somewhat, but certainly not entirely.
As a general rule, you straighten out something that is wrong or needs to be solved, settled or corrected. This might be a problem, a conflict, a dispute, a misunderstanding, a bad situation that has gotten out of hand (= out of control) etc. You can also straighten out someone who is troublesome or undisciplined. The phrase might allude to the idea of hammering straight (making straight with a hammer, or hammering out) a bent object, such as a bent metal strip — or, in other words, "straightening it out". The idea, as do a vast number of English expressions, has come to be applied in all sorts of figurative contexts. Examples:
He was willing to help his sister straighten out (= resolve) her financial affairs.
There are a few problems that need straightening out (= need solution).
I had to go back to my hometown to get some stuff straightened out.
Don't worry. Things will straighten out (= things will get better).
The phrase can also be used in its literal sense, as in "the path straightens out after the third bend" or "she straightened out her legs". These are instances where "out" is added for emphasis and can be dropped with little or no loss of meaning.
"To straighten up" is frequently used to mean "to make neat or tidy". The expression doesn't carry as strong a nuance of resolution and problem-solving as "to straighten out". You might straighten up your room because it's too messy, or straighten up the house because you have company on their way over. "To straighten up" can also mean to correct one's posture by sitting or standing up straight, and, by figurative extension, the idea of correcting one's misbehavior (e.g., "the mischievous child promised to straighten up"). "To straighten up and fly right" is a relatively well-known phrase meaning to start behaving properly after a period of misconduct.
I hope this helps.