On a simple level, the Vilna Gaon explains that, as before (Esther 5:3), a request is personal, whereas a petition is for the needs of others. After all, it is the duty of a queen to petition her king.
Similarly, the Maharal writes that a “request” is a natural desire. A “petition” is an above-average request.
Rav Moshe Dovid Valle writes that “request” can be answered immediately, but a “petition” would require more time and effort.
The Malbim writes that a “request” is just the act of asking for something. A “petition” is the providing of a reason for that request. A child can request from his father money with the petitioned purpose of acquiring some land. A father may want to hear both the plea and its reason, especially if he may want to provide both the money and the land.
The Dubno Maggid writes that this was Achashverosh’s intent – to provide Esther with both her desired goal and its means.
Perhaps, as suggested by class participant CL, Achashverosh was suspicious of Esther, and wanted to know the core intent of her request.