On the one hand, we are used to seeing university students carrying their laptops around and typing very fast pages and pages of notes. First of all, a clear advantage from typing is that it is the fastest method. Not only that but it is also more understandable and easy to read than handwriting.
Besides, some people might say that it gives the impression that you are more professional, implicated and even "updated" with technologies, and I agree with that, because, for instance, when you are submitting a CV, a project or any important document you type it, no matter how nice your handwriting is.
On the other hand, there are also many pros in having a good looking handwriting, especially at school, where sometimes laptops are not allowed and you must write down everything. In that case, handwriting is useful and should be practised, due to that it is important to write as fast as possible while keeping in mind that you have to reread it later, not an easy task! Also the way you write may influence your mark in an exam because the teacher always takes into consideration how the test is structured and if it is visually appealing, which are factors where your handwriting influences.
To sum up, I firmly believe that both methods can coexist together and that handwriting will never die. Even if school decides to allow the use of new technologies in their classes many people would still rather use paper and pen, and there's nothing wrong with that because typing doesn't have to be for everybody.