The etymology of the words "urethane" and "carbamate" are highly similar but not the same. The word "urethane" was first coined in 1833 by French chemist Jean-Baptiste Dumas. Dumas states "Urethane. The new ether, brought into contact with liquid and concentrated ammonia, exerts on this substance a reaction so strong that the mixture boils, and sometimes even produces a sort of explosion. If the ammonia is in excess, all the ether disappears. It forms ammonium hydrochlorate and a new substance endowed with interesting properties." Dumas appears to be naming this compound urethane. However, later Dumas states "While waiting for opinion to settle on the nature of this body, I propose to designate by the names of urethane and oxamethane the two materials which I have just studied, and . These names which, in my eyes, do not prejudge anything in the question of alcohol and ethers, will at least have the advantage of satisfying chemists who still refuse to accept our theory." The word urethane is derived from the words "urea" and "ether" with the suffix "-ane" as a generic chemical suffix, making it specific for the R2NC(=O)OR (R not = H) bonding structure.
The use of the word "carbamate" appears to come later only being traced back to at least 1849, in a description of Dumas's work by Henry Medlock. Medlock states "It is well known that the action of ammonia on chloro-carbonate (phosgene) of ethyl gives rise to the formation of the substance which Dumas, the discoverer, called urethane, and which we are now in the habit of considering as the ether of carbamic acid." This suggests that instead of continuing with the urethane family naming convention Dumas coined, they altered the naming convention to ethyl ether of carbamic acid. Carbamate is derived from the words "carbamide", otherwise known as urea, and "-ate" a suffix which indicates the salt or ester of an acid.
Both words have roots deriving from urea. Carbamate is less-specific because the -ate suffix is ambiguous for either the salt or ester of a carbamic acid. However, the -ate suffix is also more specific because it suggests carbamates must be derived from the acid of carbamate, or carbamic acids. Although, a urethane has the same chemical structure as a carbamate ester moiety, a urethane not derived from a carbamic acid is not a carbamate ester. In other words, any synthesis of the R2NC(=O)OR (R not = H) moiety that does not derive from carbamic acids is not a carbamate ester but instead a urethane. Furthermore, carbamate esters are urethanes but not all urethanes are carbamate esters. This further suggests that polyurethanes are not simply polycarbamate-esters because polyurethanes are not typically synthesized using carbamic acids.
IUPAC states  "The esters are often called urethanes or urethans, a usage that is strictly correct only for the ethyl esters." But also states, "An alternative term for the compounds R2NC(=O)OR (R not = H), esters of carbamic acids, R,NC(=O)OH, in strict use limited to the ethyl esters, but widely used in the general sense". IUPAC provides these statements without citation.