44 States allow lawful carry of firearms in the common terminal and baggage claim areas of airports. There are federal laws that restrict firearms, knives, and other weapons in the "sterile area" beyond the security checkpoint but the unsecured areas of the terminal are subject to state law.
Florida, and only five other states, make the entire airport terminal off limits even to carry license holders while armed. The US Supreme Court made it clear in DC v. Heller (2008) that Americans have the right to "carry weapons in case of confrontation," but this right can be restricted "in sensitive places".
The Federal government has determined that the "sterile area" is the "sensitive place" at an airport but Florida has gone much farther and banned even licensed carry inside the entire terminal, on this topic Florida is one of the most restrictive places in the country.
This has lead to confusion in the past even outside the terminal. In one case a lawful concealed carry licensee had his firearm become accidentally exposed while helping to unload luggage for someone who he was dropping off at the airport. Airport police overreacted and he ended up cuffed on the sidewalk outside until the officers realized their mistake.
These incidents don't happen in the states where the limit on carry is clearly defined as in the "sterile area".