Why do we count in two ways?

Question

To prove combinatorial identities, I've always been taught to count in two ways. Why do we do this, rather than just use algebraic manipulations to go from one side to the other?

Answer 1

Because it is fun. ${}{}{}{}{}$

Answer 2

Short Version.

Because the most interesting problems regarding counting are insanely difficult to solve using algebraic tools only. And sometimes, even with algebra as an aid they are still difficult.

Long Version

Try to count the following using only algebra. $\underline{\text{No cheating, don't look it up.}}$

Let $(a,b) \in \mathbb{Z_+}^2$, that is $(a,b)$ is a lattice point on the first quadrant including both axis. We define a step by moving one unit north, east, south, or west.

Restrict any path to $(a,b)$ to the first quadrant only. In how many ways can you get to $(a,b)$ using $a+b+6$ steps as defined above?