for me, apologies can be helpful for me to "get over myself", especially in a short term situation. and frankly, i am not one to scrutinize motive too closely because i am pretty quick to forgive in general. if they even KNOW they should apologize to me, i can accept that gesture, even if it was perhaps not as sincere as it could be.
however, it's taken me about 10 years to fully learn how to forgive someone who never apologizes or expresses remorse, or even awareness of her painfully hurtful behavior. that ability has translated to helping me not be dependent upon apologies for forgiveness.
the relationship type, relative expectations of the relationship, and how much i have invested with that person, has a lot to do with whether i would continue a relationship with someone who has hurt me, and done so without regret.
what is more important to me than the apology is the awareness and learning that comes from the resolution of the problem. if i sufficiently feel that the conflict resolution provided some means of growth and understanding, and that it will prevent future problems (potentially) then i am more likely to give more chances.
a lack of awareness, understanding or responsibility leads me to be concerned whether this will be a repetitive cycle of offenses and inability to accept/learn from and nurture the growth of the relationship.