Yes, you did something wrong.
First of all, when you purchase guns like that you have to act as a manager; laying out the steps of the process, making sure everyone has the correct information, and following up.
The FFLs only need to exchange signed copies of their FFL licenses. This is mandated by law and is a completely separate arrangement from your arrangement to purchase a gun.
*YOU* were supposed to send the payment to the dealer.
How did you pay? However you paid, you'll have to verify that it went through. However, you still have a problem. Since you depended on your dealer to send the payment, it's possible that he just stole it for himself.
It may also be possible that the dealer received the payment with the license, but didn't make the association of that payment to your purchase (it may be in his find-out-what-this-is-about pile). Call and ask if he received the license from whoever your FFL is. Inform him that your payment was with the license.
You're going to have to do some investigating to straighten out this mess.
Here's how one usually goes about it.
Agree to buy the gun.
Send dealer the payment and the name, address, and phone number of the transfer dealer.
Call dealer to ask if he contacted transfer dealer.
Call transfer dealer a day after he has been contacted to see if he mailed out his FFL.
A week after transfer dealer mails FFL, call dealer to see if he got it.
After the dealer receives the FFL he should ship the gun.
You should be able to transact the purchase in less than two weeks if you stay on top of everyone.