What if the Gulf of Mexico didn't exist?

It is barely possible based on considering the relative depths of the Gulf and the Caribbean.

If it were that way with the Earth otherwise maintaining its current shape, it would imply a much lower ocean level (sea level) and a world with radically less water.  We would have fewer fish because the oceans would be smaller.  With less water would come less arability of the land, meaning our populations would have to be rigidly controlled or we would have mass starvation.

On Earth, we have a rather dramatic transition from gas (our atmosphere) to solid (the Earth's crust). That sharp transition creates a medium in which we can easily move through the air but receive enough resistance from the solid that it can support us - we can stand on it.

On Jupiter there is no such transition. Instead as we work our way down through Jupiter's atmosphere it just keeps getting denser and denser. Eventually the gaseous hydrogen becomes liquid hydrogen. We keep descending and it becomes denser and denser until that liquid hydrogen becomes a super hot liquid metal. If we kept descending, we likely would eventually come to a solid core. But the interior pressure on Jupiter is millions of times greater than on the Earth. The heat and pressure would have destroyed us and our spacecraft long before we got to that core. The extreme density of the liquid would have made it impossible for us to even propel ourselves towards the core.

The GULF Stream is warmed up by the time the water is in the gulf.   No gulf then no warming.   Europe would be  LOT colder than it is,   North Africa would probably get more rain and not be a desert and the Carib would only be hit from the east with hurricanes.   

The drive from Tampa to Corpus Christi would be shorter, you would skip Mobile, New Orleans, Houston.  The Mississippi would be longer.  Mexico would have more land area.  Offshore oil drilling would be on shore.  The Gulf Stream flow might not exist, and a lot of Europe would freeze solid in winter.

The Gulf of Mexico didn't exist as such during the Paleozoic Mesozoic eras. 

The meaning of your question is unclear.  Are you saying there would be swampland (or what) in the area where the Gulf of Mexico is now found?