"First use" determines the moment when liability for the tax arises. Form 2290 must be filed by the last day of the month following the month of first use. The Form 2290 Instructions show how to determine the due date for filing Form 2290.
"Purchase date" is used in the context of registering the vehicle, not liability for tax. It is important because under current regulations, within 60 days of purchase, some states allow the registrant to register with the bill of sale instead of the stamped Form 2290, Schedule 1. Generally, Schedule 1 is required as proof of payment of tax for purposes of registering the vehicle with the state. Check with your state to find out if it allows registering with the bill of sale.