Hi there, business analyst and associate is largely a similar role. In general, you are the content leader, ie nobody knows the specific client situation and topic better than you, not you manager, not your partner. However, there is different level of quality and robustness required when comparing BAs and Associates.
I will take a shot at 3 key differences of expectations at problem solving:
1) independence - as BA you could be guided, as an Associate you develop ideas and solutions indendependently and contribute from Minute 1 (you can do that as BA as well, but as an Associate thats a must)
2) ability to make sound decisions - you are navigating ambiguity constantly, you need to be comfortable making backed up decisions and fast as an Associate, you can get away with little bit of hesitation and insecurity as a BA
3) robustness of your ideas - as an Associate you have either been at the Firm for 2-3 years or came from intereting industry/vertical hence you come up with out-of-the-box solutions, whereas as a BA you are likely coming out of school and again, you can get away with less depth, given your professional experience beforehand
I am sure there is a tonne of other differences and best BAs can outshine poor performing Associates, but these 3 areas are in my view key differentiating factor, given similarity of the roles.