Are the parties both women? It's just that "maiden name" is only used of women (presumably since men can't be maidens!):

"A "maiden name" is the family name carried by a woman before any of her marriages... A married name is a name (if) changed by a married woman."
http://en.wikipedia.org/wiki/Maiden_name

If we're talking about a man and a woman, however, "maiden name" would not be appropriate. In that case you could use "birth name" instead:

"The term is ambiguous for those who changed their birth name before any marriages; therefore some prefer the term "birth name", which can also be used in the case of a man changing his name upon marriage."
http://en.wikipedia.org/wiki/Maiden_name

"Continue to use" would be okay, I think, but the usual way of saying it (and more appropriate in a legal context) is "retain".

"As a result of the ambiguity in current Illinois law, a married woman who chooses to *retain her maiden name* may face inconvenience and possibly serious legal consequences."
http://www.cwluherstory.com/CWLUArchive/name.html

"A wife usually assumes her husband’s last name. However, she is under no legal compulsion to do so, and she may retain her maiden name at the time of marriage with no formal legal proceedings."
http://www.unc.edu/student/orgs/sls/pages/topics/marriage.ht...