First of all, let’s ignore that it’s cappuccino; cappuccino is espresso with foamed and steamed milk, and milk doesn’t have caffeine (if your milk has caffeine, you have some very unusual cows!). So we’re comparing espresso and coffee. Espresso is made by forcing steam through fine coffee grounds, extracting a great deal of the aromatic oils, VOCs, and caffeine from the roasted coffee.
When you say “coffee”, you probably mean “drip coffee”, which is made by – as the name implies – allowing hot water to drip through medium-ground coffee. It’s nowhere near as efficient an extraction as espresso BUT we tend to drink vastly more of the stuff.
Yet, that hits our first major bump. Quantity. A “cup of coffee” can run anywhere from 4 oz (a typical, albeit small, coffee cup) to 20 oz (Starbucks venti). Espresso is usually served as 1 oz shots, and a cappuccino may contain one, two, or even three oz. So that’s the biggest question: how much of it are you drinking? 20 oz of drip coffee will likely have more caffeine in it than 1 oz of espresso; 3 oz of espresso will almost certainly have more caffeine than 4 oz of drip coffee.
Another factor in the amount of caffeine is the bean or blend. Various varietals of coffee bean have more or less caffeine. Premium (i.e., what people like me consider “good”) coffee will be Arabica; the contrasts with the Robusta bean. Robusta has considerably more caffeine than Arabica. Exact amounts vary, but the more Robusta bean, the more caffeine compared to Arabica.
Now, you might be inclined to believe that the roast of the bean is important. After all, that really dark Seattle roast tastes stronger, doesn’t it? It does matter, but not directly. The roasting has a trivial effect on the caffeine content of the bean BUT it affects the size of the bean. During roasting, the coffee bean “pops” and gets bigger. The more you roast it, the bigger it gets (simplifying a bit on this). The caffeine content stays the same, the mass stays mostly the same, but the volume increases. What this means is that if you measure your coffee by *volume*, there’s more caffeine in a lighter roast coffee. If you measure your coffee by *mass*, there’s more in the darker roast. However, either way you look at it, the difference is pretty trivial.
Now, what should be reasonably clear is that assuming you’re using the same coffee beans, same roast, and same measuring method, espresso will have more caffeine by volume than drip coffee. It’s just a more effective extraction method – more pressure, higher temperature, greater surface area (due to the finer grind). Espresso extracts more.
BUT you asked about cappuccino. Unlike espresso, we tend to consume cappuccino in roughly the same volume as drip coffee: we make up the difference with the aforementioned milk. So it goes back to how much espresso was put in your coffee. In general, a 1 oz shot of espresso has a little less caffeine than a 4 oz cup of coffee, so if your cappuccino has just one shot of espresso it likely has less caffeine by volume; if it has two or more shots, it has more.
I’m afraid there’s one other twist, though. Even if your cappuccino has less caffeine than drip coffee, it still probably has more calories than drip coffee. A cappuccino has quite a bit of milk in it, more than you could conceivably be adding as creamer. And it likely has one or more pumps of flavored syrup – each of which are roughly equivalent to a tablespoon of sugar. So, unless you’re putting a huge amount of sugar and cream in your coffee, a cappuccino far outpaces drip coffee on calories. Those, in concert with the caffeine, will give you more energy (or make you more jittery, your mileage may vary) than the same amount of drip coffee.