Many European butters are identifiable by the food the cows eat (for example, they only eat grass on particular cliff by the sea in a particular province in a specific country), and most American butters (the mass-produced ones, not the artisan ones) are a blend of milk from no identifiable source. Often the European butters taste so good because the grass and feed are so specially sourced and consistent.
European butters, as others have correctly said, contain more butterfat. They are better for baking and for spreading on good quality baked goods (like an excellent croissant, but not an ordinary piece of toast, for example), but they are not acceptable for use in sauteeing, due to the increased fat content (they tend to burn faster).
I took a cooking class given by a chef at one of the most prestigious 5 star hotels in the country. He said: use European butter in baked goods (less water, more fat), and when serving fine pastries or homemade special breads. He also said that the only American butter he would allow in his kitchen, for routine uses, was Challenge Butter - their traditional style. (Challenge makes a traditional butter and a European-style higher fat version as well). He preferred KerryGold for his baked goods and special presentations.
Cultured butter is different altogether. Some European butters are cultured or have cultured butter added to their formula, but not all cultured butters are European. A cultured butter is made from properly aged milk or cream and bacteria (the good kind) are allowed to slowly ferment. Then the cultured cream or milk is turned into butter. It has an even higher fat content and it's delicious. Interesting thing: people who are lactose-intolerant (NOT allergic to dairy: that's an entirely different situation) can sometimes enjoy cultured butter, because the bacteria have consumed the lactose that bothers the lactose-intolerant person.