Just looking at a map would explain the high temperature of the GOM. Most of its water never leaves but just circulates over and over in the same basin between Texas and Florida.  With winter temps not being as low as 20-30 years ago the summer heating just carries over to the following spring.
As for any connection to the Greenland Sea I think some of you confused the Gulf Stream with the Gulf of Mexico. The two are on opposite sides of Florida.
Actually, the Gulf Stream does flow out of the Gulf of Mexico. It flows through the Straits of Florida north of Cuba.  So slower sinking of surface water off of Greenland could slow the exit of surface water from the Gulf of Mexico, thereby allowing its heat to increase.