My interpretation of each expression in the absence of any further elaboration would be as follows. Let's suppose today is Thursday.
Last week, I would take to mean, sometime between Monday and Friday of the previous week. If it had happened at the weekend, I would expect the speaker to say at the weekend, or the weekend before last. Though that might not always be the case.
A week ago, would suggest to me, since today is Thursday, that it happened roughly in the range Wednesday to Friday of the previous week.
But a lot of this could depend on the context of the conversation, to whom the person was speaking etc. Were it their insurance company's claims department it could perhaps be interpreted more literally than if it was a casual conversation to an acquaintance in a pub.