Yes, ‘respect’ , often the plural ‘respects’, refers to the ways in which you see something or deal with something : ‘the two situations differ in many respects’ , ‘the two [people] are similar in many respects’. ‘In this one respect the situation is different to that we dealt with before’. The situation or the person has characteristics that can be identified by us.
‘Aspect’ also refers to the way you look at or perceive something, but is more definitive : ‘there are aspects of the situation that have not yet been addressed’ , ‘if we look at it from that aspect, it seems a lot better’, ‘there are some aspects of his character I find confusing’. It is our perception of the situation, person, etc that defines these aspects.
In some cases the two are interchangeable : ‘from some aspects the situation appears manageable’ , ‘in some respects the situation appears manageable’. You can see the distinction here : aspects are the ways you look at the problem, respects are the intrinsic characteristics or properties of the problem. In practice one or the other word is usually more appropriate, but it really doesn’t matter that much !