The snapping turtles are members of the New World turtle family.

According Mlynarski's classification, there are only two full species in the snapping turtles:
1. Chelydridae family: Chelydra serpentina, the common snapper,
with only four currently recognized subspecies
2. Macroclemys Temmincki , the alligator snapper, with only one subspecies in this group.


The full classification is as follows (Mlynarski, 1969):
Order: Testudines
Suborder : Cryptodira
Superfamily: Chelydroidea
Families:
1. Chelydridae
1.1. Chelydra serpentina serpentina - The Northern common snapping turtle - the Common Snapping Turtle (Linneaeus 1758)
1.2. Chelydra serpentina osceola - the Florida Snapping Turtle (Stejneger 1818)
1.3. Chelydra serpentina rossignoni - the Mexican Snapping Turtle (Bocourt 1868)
1.4. Chelydra serpentina acutirostris - the South American (Equadorian) Snapping Turtle (Peters 1862)
2. Macroclemys
2.1. Macroclemys temmincki - the Alligator Snapping Turtle (Troost 1835)

(Source: http://www.chelydra.org/snapping_turtle_classification.html)
Common snapping turtle vs Alligator snapping turtle
(Source: http://www.chelydra.org/common_alligator_snapping_turtle.html)

 Common Snapping Turtle vs. Florida Snapping Turtle
(Source: https://backwaterreptilesblog.com/common-snapping-turtle-vs-florida-snapping-turtle)

 Both species of snapping turtle have soft spikes on their necks and heads that are known as tubercles. Common snappers have rounded tubercles, whereas the tubercles on Florida snappers are pointed.
The second physical difference between the two species of snapper is coloration. Florida snappers tend to be a warm, light brown in color when they are young. Common snapping turtles, on the other hand, are usually a dark grey or black.