Mid-July is horrific. June seems too damn hot until you're in July and you remember how hellish it is August can be even worse.
June is still often quite hot (can hit 100F in inland areas away from the beach) but in most years, it's a little bit drier/lower in humidity, and that lower dew point does matter in terms of comfort.
In my opinion, it is all about water temperatures. In June, the Gulf will not be as warm as in July and certainly not August. I am not implying the Gulf is cold in June, but it is still cooler. The reason October is so liked by locals is the air temperature is often cooler with less humidity Also, less likely to have a tropical storm in June.
