Normally, it is experience and roles that differentiate an Associate Software Engineer from a Software Engineer.
Associate Software Engineer:
new person in this IT field.
Has lesser experience that may be from fresh graduates.
He does the simple jobs and learns from other engineers who have much experience.
Mostly, he learns to code and how to create software.
Software Engineer:
He has more experience and skills.
He does tasks and projects of complexity.
Will have greater responsibility, and be able to lead projects and teams.
Should be able to work more independently, solve more problems with less supervision, etc.
In other words, an Associate Software Engineer will typically be at a more junior level compared to the position of Software Engineer which will have already established some level of experience and responsibility.