I like the feeling of the explanations. However, a very obvious difference could be the quantity of time.
If today is Monday, last week could refer to something that happened on Friday, for example. That is 3 days ago.
One week ago for me = 7 days ago and is, therefore, more specific.
Then we could turn to the feelings of what is definite or indefinite, or consider that since 'a week ago' is more specific, it will be rarer.
A look in the iWeb corpus instances:
A WEEK AGO 37,053
LAST WEEK 631,873
Because the latter is more general, it will be used more often.