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Liquor vs. Liqueur
All liqueurs are liquor, but not all liquor is liqueurs.
A liqueur is defined as a spirit with a minimum ABV content of 15% and a minimum sugar content dependent on the type. Liqueurs must be made with "only natural flavouring substances and preparations." 1
Liquor on the other hand refers to a distilled spirit usually made from a base of fruits, grains, or sugarcane. Liqueurs use a liquor as a base spirit. Food & Wine has an interesting article on the best uses for liquors and liqueurs.
Liqueur vs. Crème
According to Culinary Lore: "Crème is the name given to those liqueurs that are extra sweet and usually syrupy. They have up to twice as much sugar as other liqueurs and are heavier and denser in texture. They can be sipped alone but they are also great for adding a lot of sweet flavor to mixed drinks."
Therefore main difference between a liqueur and a crème liqueur is the amount of sugar involved in the preparation. According to the EU regulations defining them:
Crème de Cassis
Crème de Cassis de Dijon like Jules Theuriet by Briottet is even "more regulated, limited geographically and must have minimum of 400g of sugar per litre compared to standard Crème de Cassis. Crème de Cassis de Dijon must also contain at least 25% of the Noir de Bourgogne blackcurrant variety." 2
Crème de Cassis de Saintonge like Jahiot is also geographically-limited, must have a minimum sugar content of 450 grams per liter and a minimum alcohol content of 18% under the IGP (l'indication géographique protegée) issued in 2014. They use primarily the Noir de Bourgogne and Royal de Naples blackcurrant varieties. 3
Crème vs. Cream
Crème liqueurs don't actually contain dairy products. Instead, the French word crème (which does actually mean cream), is used to indicate the sweet, syrupy quality of the crème.
Cream liqueurs on the other hand, do actually contain diary cream (think Bailey's Irish Cream). They are liqueurs to which cream has been added.