In other words, why didn't NASA make their main space launch area in Texas? It has a point below the 30th parallel, which means that it is close enough to the equator. Why did NASA pick Florida?
Basically, the most prominent reason is so that if something happens during launch, it happens over the Atlantic and not someone else.
Anything launching over the Gulf of Mexico will probably cross over land a couple of times before going over the Atlantic. As geoffc pointed out in the comments, the Atlantic is a lot wider than the Gulf. Once the rocket has crossed the Gulf it will still be low enough that anything that falls over land could still cause damage. Once the rocket has crossed the Atlantic, it should already be high enough that this is not really much of an issue (stuff burns up in the atmosphere as it falls).
Until 1949, the U.S. launched rockets from Wallops Island in Virginia and the White Sands Missile Range (WSMR) in New Mexico. The Rockets launched from Wallops were of American origin while the rockets launched from White Sands were V-2s, supported by a hundred or so German rocket scientists who had been smuggled out of Germany (along with some V-2s) via Operation Paperclip.
Then this happened:
The most spectacular flight in the annals of WSMR occurred May 29, 1947, when an experimental V-2, weighing four and a half tons, headed south after liftoff instead of north and landed, some five minutes later, a mile and a half south of Juarez, Mexico. Though no damage was done, the rocket narrowly missed an ammunition dump where Mexican mining companies stored powder and dynamite.
Raining rockets down on U.S. civilians is one thing. Raining them down on other countries is quite another. There were a number of near misses prior to that incident. A search for a better launch site was already underway.
In October, 1946 the Joint Research and Development Board under the Joint Chiefs of Staff established the Committee on the Long Range Proving Ground to analyze possible locations for a new missile range to be shared by the various branches of the military.
Three potential sites emerged. One was based on the coast of northern Washington, with a range along the Aleutian Islands of Alaska. A second was based at El Centro, California, with a range along the coast of Baja, Mexico. A third was based at the Banana River Naval Air Station, with launches from Cape Canaveral and a range over the Atlantic Ocean.
In September, 1947 the Committee on the Long Range Proving Ground announced its decision to recommend the establishment of a missile proving ground at the California site, with Cape Canaveral offered as the second choice. The Washington site had been quickly rejected due to its isolation and poor weather.
The project was officially designated the Joint Long Range Proving Ground, with development responsibility granted to the Joint Long Range Proving Ground Group. Although plans continued initially for the establishment of a missile range based in California, political problems arose in 1948.
Although it would have been a suitable site very close to existing missile manufacturers, the California site had to be rejected when Mexican President Aleman refused to agree to allow missiles to fly over the Baja region. This was largely a result of bad timing, since a wayward V-2 rocket launched from White Sands, New Mexico had recently crashed near Juarez, Mexico.
The British, however, were quick to express their willingness to allow missiles to fly over the Bahamas. They also were willing to lease island land to the U.S. military for the establishment of tracking stations. That, coupled with inherent strengths of Cape Canaveral, sealed its selection as the first U.S. long range missile proving ground.
Shortly after this selection, those German rocket scientists were moved from Fort Bliss to the Redstone Arsenal near Huntsville, Alabama. Ten years later, those German rocket scientists, along with thousands of American rocket scientists would be transferred once again, but this time the transfer distance was short. NASA's Marshall Space Center is located within Redstone Arsenal.
If launching into a low-inclination orbit, you want to launch due east from the lowest latitude possible. This gives you the advantage of starting the flight with the speed of the Earth's rotation at your launch latitude; that is "free" speed that the rocket does not need to impart. Furthermore, you want to avoid overflying land masses so malfunctioning rockets to do not land on people's heads, which can have negative budget implications. When launching into higher inclination orbits you adjust your launch azimuth accordingly. Your launch azimuth determines your ground track and thus which land masses you might hit with rocket debris. Typically the launch azimuth for an orbital flight is going to be between due North and due South, on the right-hand side of the compass (0 to +180 deg). Launching to the east from the Texas coast would result in overlying lots of land mass (US, Canada, Cuba, Mexico, S.America). Launching from Florida mitigates this for easterly launches. Polar orbit launches are commonly done from Vandenberg AFB in CA, flying south over the Pacific. High-inclination and polar launches are conducted from Wallops VA. Johnson Space Center's Houston location is a result of politics (President Johnson was from Texas).
I agree with the other answers that point out that there is no land east of Florida, while there is significant populated land east of Texas. But I think it is work pointing out that Texas has no advantage over Florida in terms of latitude.
The southernmost point of Texas is Brownsville (Lat 26 deg N) Which by the way, shares a border with Mexico, with which the USA sometimes has less than perfect relations. On the one hand the USA might feel there are some security issues. On the other, Mexico would be quite justified in complaining if an out of control rocket landed in their territory. Houston / Johnson is at Lat 29.5 deg N.
Canaveral / Kennedy is at Lat 28.5 deg N. The southermost tip of Florida is at 25 deg N, but is very close to the population of Miami and the sovereign nation of the Bahamas.
It is worth checking the advantage that would be obtained by launching from the southern tip of Florida instead of the existing site:
Equatorial peripheral speed: 40000km / 24h = 1666km/h, 0.462km/s
Peripheral Speed at Canaveral 28.5 deg: 1666 cos 28.5 = 1464km/h, 0.405km/s
Peripheral Speed at 25 deg: 1666 cos 25 = 1509km/h, 0.419km/s
In summary the lowest latitude point on the US mainland is in Florida, not Texas and the advantage compared with the current location would be just 45km/h (28mph.)
Hawaii's big island has a latitude ranging from 19 to 20 degrees, but again the advantage is small and probably does not outweigh the logistical issues.
Besides these other answers, there was a great post-WW2 effort to develop the American South.
So the space program was spread between the southern states, e.g, New Mexico, Texas and Florida.
So the command centre is in Texas, the launch site in Florida and the landing strips in Florida and New Mexico, among other sites.
As many others have stated Florida is a great place to launch rockets because if the rocket suffers an issue during launch debris from it will land in the Atlantic ocean rather than crashing down into inhabited land, possibly damaging property or killing civilians. Another reason that i don't think has been brought up but bears mentioning is that in 1949 swampland in Florida was considerably cheaper to buy in bulk than the alternative sites they were considering.
Source: Heard this on tour of Kennedy Space Centre.