There are two significant factors in answering this question. First, I would argue that the culture of Florida is historically Spanish when compared with the rest of the Deep South. Critically important to the development of Florida as an American state is it's late addition to the Union. Florida was purchased from the Spanish in the Adams-Onis Treaty of 1819 largely because Spanish possession of Florida was destabilizing to the plantation economies of the surrounding southern states. Though slavery was legal in the Spanish Empire and existed in Spanish Florida, during the eighteenth century Spain relied on freed Africans and Native people of the region to defend Spanish positions on the peninsula. Furthermore, in order to destabilize British colonies to the north, Spain offered a freedom to escaped slaves from the American South who managed to reach key destinations like St. Augustine. Such opportunities inspired rebellions including the famous 1739 Stono Rebellion in South Carolina. During the period of Spanish control, Florida, along with other areas along the Gulf Coast, was more an extension of the Caribbean than it was a part of the American mainland. Florida more closely resembled Spanish Cuba and British Barbados with its attempts at sugar plantations, though the Florida soil was too marshy to develop any profitable large scale plantations like those of the islands. After the Anglo-Spanish War (1762-1763 as a brief period in the Seven Years War) Florida was officially transferred to British control. Although no Spanish residents were required to leave the territory, there was a large exodus of Spanish Catholics who feared reprisals from the British government. Very few British moved into the region during the colonial period and although the territory remained loyal to the British crown during the War for Independence, the region was transferred back to Spanish control in 1783 with the Treaty of Paris.
During the War of 1812, Americans became increasingly wary of a hostile Native American and free African presence along their southern border. During the First Seminole War, Andrew Jackson led operations into West and Spanish Florida in an attempt to stamp out Native-African-Spanish resistance. The weakening hold of the Spanish Empire over the their territories in the New World led the Spanish to sell Florida to the American government lest they lose it outright to military conquest. Over the next century the American government radically reshaped Florida by the forced removal of people who shared Native American and African ancestry. Florida joined the union at the time when the industrial demand for cotton made it the most lucrative export from the southern states. However, Florida's climate and soil made cotton production impossible through most of the state, the exception being in the western panhandle. Because of this, the large cotton plantations that dominated states like Georgia, Mississippi, Alabama, and Louisiana never developed in Florida. However, smaller plantations and slaveholding still provided the dominant economic model in Florida. Though Florida seceded with the other Confederates states, there was relatively little fighting in the state and the majority of the sea ports remained under Union control throughout the war.
Secondly, costal Florida has become a destination for northerners to retire. As to why parts of Florida might appear to be more northern today, especially along the Atlantic and Gulf coast, it is imperative to recognize the influence of northern retirees that began making the region popular. Beginning in the late 1960s and 70s retirees from the American Northeast, and to a lesser extent eastern Canada, began to purchase vacation homes along the Florida coast where the winter climate was more desirable. Although today prices of homes in some of these developments can be in the millions of dollars, when the boom first began land was plenty and labor was cheap creating an enticing opportunity for northern development companies to buy up the land.
TL;DR Summary: Florida remained a Spanish possession until 1819 and never developed the same cotton economy that dominated the American South in the ninteenth century. In the twentieth century the state became a destination for northern retirees.
Sources:
Jane Landers, Black Society in Spanish Florida
Peter Wood, Black Majority
Walter Johnson, River of Dark Dreams
Paul Pressly, On the Rim of the Caribbean
As /u/pickin_grinnin notes, Florida has been a popular place for northerners to retire for decades, either permanently or seasonally (snowbirds).
There's also the fact that Florida's settlement by Americans is more recent than the rest of the South except for Texas (where it was vaguely concurrent), and Florida's population growth was initially lower than the rest of the South - it had by far the lowest population in the South in 1900 (by a large margin), for example:
| State | 1900 pop. |
|---|---|
| Florida | 528,542 |
| Oklahoma | 790,391 |
| South Carolina | 1,340,316 |
| Louisiana | 1,381,625 |
| Mississippi | 1,551,270 |
| Alabama | 1,828,697 |
| Virginia | 1,854,184 |
| North Carolina | 1,893,810 |
| Tennessee | 2,020,616 |
| Kentucky | 2,147,174 |
| Georgia | 2,216,331 |
| Texas | 3,048,710 |
| Missouri | 3,106,665 |
By 1950, we see Florida catching up, but still 8th of 13 Southern states:
| State | 1950 pop. |
|---|---|
| South Carolina | 2,117,027 |
| Mississippi | 2,178,914 |
| Oklahoma | 2,233,351 |
| Louisiana | 2,683,516 |
| Florida | 2,771,305 |
| Kentucky | 2,944,806 |
| Alabama | 3,061,743 |
| Tennessee | 3,291,718 |
| Virginia | 3,318,680 |
| Georgia | 3,444,578 |
| Missouri | 3,954,653 |
| North Carolina | 4,061,929 |
| Texas | 7,711,194 |
It's now #3 in population for the entire country with 20+ million people and #2 in the South (behind Texas), mostly due to internal migration and immigration. (A casual look at /r/FloridaMan shows that their population growth is not due to having boatloads of genius children)
That said, Florida's demographics and culture is highly regional, with South Florida trending much more Latino (notably Cubano), the Panhandle and Northern Florida resembling the traditional South, and Central Florida resembling more of a classical melting pot (with the coasts having a high concentration of retirees).