Quantum computers and classical computers differ in their fundamental operation and capabilities:

Classical computers operate using bits (0s and 1s) and follow Boolean logic. Each bit must be in a definite state - either 0 or 1. These computers excel at tasks like arithmetic, data processing, and running deterministic algorithms.

Quantum computers, on the other hand, use quantum bits or "qubits." These leverage two key quantum mechanical properties:

1. Superposition: Qubits can exist in multiple states simultaneously (both 0 and 1 at once)
2. Entanglement: Qubits can be connected in ways that the state of one instantly affects another, regardless of distance

These properties give quantum computers unique advantages for specific problems.

A prime example is database searching using Grover's algorithm. Consider searching an unsorted database with N entries:

- A classical computer would need to check an average of N/2 entries (and up to N in worst case)
- A quantum computer using Grover's algorithm can find the answer in approximately √N steps

For a database with 1 million entries, a classical computer might need 500,000 operations on average, while a quantum computer would need only about 1,000 operations - a dramatic speedup.

This quantum advantage scales even more impressively with larger datasets, though it's important to note that quantum computers won't be faster for all types of problems. They offer specific advantages for problems involving searching, optimization, simulation of quantum systems, and certain types of factorization.