The difference is largely a product of the fact that a larger share of voters in much of the Northeast are tenants relative to much of the rest of the United States, primarily due to low rates of home ownership in places like New York City (46% of all New York State residents rent, the highest in the U.S., the percentage is much higher in its major cities, and municipal level rules are the source of most tenant protections in that city).
The high rates of renting there is largely due to a combination of high density, together with the fact that the development of the condominium system of ownership for multifamily dwellings (first used in the U.S. in Utah in 1960) after legal arrangements for residential real estate were already in place for most residential real estate there. In places where multifamily buildings were developed more recently, many multi-family buildings are organized as condominiums rather than as rental apartments.
Thus, greater protection for renters is really mostly a product of the median voter theorem in municipal governments and the great power given to municipal governments in those states.
Florida also has a lot of renters compared to the national average, and importantly, has a lot of politically powerful urban migrants from the Northeast, who bring their political and legal expectations with them.
The willingness of attorneys to get involved on behalf of tenants in Florida but not Texas probably also reflects the fact that there are more affluent tenants in Florida (often retirees) who can afford to pay an attorney's bills, than in Texas, where housing costs are low and renters are less likely to be affluent.