Is there a difference between a cougar, a puma, a mountain lion and a panther?

The cougar is known by many names including mountain lion, puma, cougar and panther. These names all have different origins. The name mountain lion comes from the early Spanish explorers who called it ‘gato monte’ which translates to ‘cat of the mountain.’ The name ‘panther’ is a generalised term which refers to a cat with a solid coloured coat which they have. The name cougar comes from an old word that South American Indians had which was, cuguacuarana which got shortened to cuguar and was then spelled differently. Puma is the commonly accepted scientific term.

http://theanimalfacts.com/mammals/mountain-lion-cougar/

Despite their large size, the cougar is not a true “big cat”. The cougar is a member of the felidae family, as opposed to the panthera family as of the tiger and lion. The major difference between the two families is the panthera family’s ability to roar.

Felidae cats can not roar, instead make noises such as purrs, hisses, chirps and screams.

"The cougar has numerous names in English, of which puma and mountain lion are popular. Other names include catamount, panther, painter because of its black tail tip, and mountain screamer. In North America, "panther" is used most often to refer the Florida panther sub-population. In South America, "panther" refers to both the spotted and black color morphs of the jaguar, while it is also broadly used to refer to the Old World leopard.

The cougar holds the world record for the animal with the highest number of names due to its wide distribution across North and South America. It has over 40 names in English alone.

"Cougar" is borrowed from the Portuguese çuçuarana, via French; the term was originally derived from the Tupi language. A current form in Brazil is suçuarana. "Puma" comes, via Spanish, from the Quechua language of Peru."
Source and further information:
https://en.wikipedia.org/wiki/Cougar

Some other names:
"The names "Erie" and "Eriez" are shortened forms of "Erielhonan," meaning "long tail." The Erielhonan were also called the "Cat" or the "Raccoon" people."
https://en.wikipedia.org/wiki/Erielhonan

"In North America, the Spanish call the lion léon de montaña, the Mayans cabcoh and the Guianans tig rouge. Texans also have names for their lions, including puma as in its taxonomic designation Puma concolor, cougar, panther, and catamount, meaning “cat of the mountain.”"
http://www.tpwmagazine.com/archive/2005/jun/legend/

Don't forget the Nittany Lions that once proudly roamed the hills of central Pennsylvania.

Mountain lions had roamed on nearby Mount Nittany until the 1880s, when it is believed they were eliminated by fraternity initation practices in the late 1800s. Students wore the pelts of the captured beasts and some are still in use in private pledging rites by some of the older Penn State Amicae societies. Though anecdotal sightings persist, it is believed that these are pledges bound in the rotting pelts of the last of the wild lions.

https://en.wikipedia.org/wiki/Nittany_Lion