What are the benefits of quantum computing over classical computing?

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Benefits of quantum computing are mostly in speed which quantum algorithms have over classical ones. 

Quantum algorithms are those which use some features of quantum mechanics such as quantum entanglement, quantum superposition and Heisenberg uncertainty principle. So quantum computers allow us to implement faster and more efficient algorithms, they allow us to use a different way of thinking.

Some of the most famous quantum algorithms is Shor's algorithm for integer factorization which works in polynomial time, `O((log (n))^3)` to be exact, while the fastest classical integer factorization algorithm works in super-polynomial time (slower)
 `O(e^[1.9[log(n)]^(1/3)cdot[log log n]^((2)/(3))])`
There is an interesting  consequence of Shor's algorithm. Because Shor's algorithm is much faster than than classical one and since many algorithms in cryptography use integer factorization, quantum computer would be able to crack such cryptographic security much faster than classical computer.
One other quantum algorithm is called Grover's algorithm for searching of unordered list and it takes `O(sqrt(n))` time while the classical algorithms work in `Omega(n)` time which is much slower.
I have said Shor's algorithm can crack classical cryptography which uses integer factorization quite easily, but that doesn't mean that there would be no data security in world of quantum computers. Quite the opposite there are quantum cryptography protocols such as BB84and B92 which are much safer than cryptography protocols used today.

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