There are many differences, one of them is that when you buy a calculator it will serve you well. Other is that calculator does not confabulate the result from time to time and you don't have to pay attention and recheck everything it calculated... And the calculator does not provide you with the most appealing results that are _de facto_ calculated to deceive you
Small update to the differences here: calculator won't ever need a GPU or other device that cost is 50k$ and you'd always need to rent it for seconds. Calculator also won't be embargoed at any point of time (i.e. when next USA president comes into the office, he/she won't be able to prevent your further usage of calculators in your country). Also all the things done by calculator you could do on the paper, just slower.
The integration calculus is a little bit different and more connected to programming - you need to work through as many possible types of integrals as possible to get those good methods of solving into your mind, and then exercise them through and through to get them into your long lasting memory. The same in programming patterns, ways of solving problems etc.