I am answering here as it applies to Italy, elsewhere it can be different.
Cappuccino in Italian means little cappuccio, which in turn means little cap. It is a milk-and-coffee drink with a peculiar topping: a milk foam obtained with steam. You might ask to add also some chocolate powder. Apart from the taste, the powder filling the foam pore gives a pleasant look to the drink. When travelling abroad, I see this as a standard feature, but here it is on request.
The steam is produced from a pump part of quite a large coffee machine, which is the standard equipment of any Italian bar. Because of the high pressure involved it is normally not possible to produce a cappuccino at home, at least not using a standard stove. Anyway nowadays they are selling cut down versions of the bar machine, which can produce bar-like cappuccinos (and coffee) even at home.
Temperature is crucial for cappuccino. Hot temperature is necessary to produce the foam, but a too hot cappuccino will hurt your tongue. Contrary to a hot cup of tea, where you can wait for it to become colder, here when the hot goes away also the foam does, so what you drink is not a cappuccino anymore.
Espresso means express, as you might guess. It is a coffee made with the big machine I mentioned above. Contrary to the coffee made at home on the stove, with the so called moka machine, this machine can make a good coffee instantly. That anyway requires this appliance to be already setup (with proper pressure and temperature levels), which takes time but it is done only once, in the morning, when the bar opens. it might help observing that breakfast is almost inexistent in Italy, as it consists for many of a fast and frugal espresso or cappuccino with a croissant, taken standing still at the bar before commuting to you workplace. So speed in brewing pays.
Besides Italian approach to coffee as a drink is radically different. For what I have seen, in the rest of the world the coffee is drunk in mugs similar to those used for the tea (I am thinking to American coffee for example); here the size of a coffee cup is extremely small and this is because the drink is concentrated. To brew the coffee like this you need water at high pressure and high temperature forced through a tier of coffee by the coffee machine.
Big espresso machines found in bars are ideal, given their power, to obtain this and generally their output is considered better than moka coffee made at home.
Note that contrary to cappuccino, here the higher the temperature the better the taste. To make it drinkable bars use special cups very thick. The final results is that you don’t hurt your lips when you drink, also the volume of liquid going through your mouth is less, and you don’t burn your tongue.
I have read that the total caffeine in an American coffee is higher, anyway the sensation that you have with an espresso is like you are taking something stronger and we even amplify this effect by drinking it fast, often in one shot.
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