I assumed that we work in a real inner product space, otherwise of course we have to put the modulus.
The inequality $\langle u,v\rangle\leq \lVert u\rVert\lVert v\rVert$ is also true, but doesn't give any information if $\langle u,v\rangle\leq 0$, since in this case it's true, and just the trivial fact that a non-negative number is greater than a non-positive one. What is not trivial is that $\lVert u\rVert\lVert v\rVert$ is greater than the absolute value. But in fact the assertions
$$\forall u,v \quad \langle u,v\rangle\leq \lVert u\rVert\lVert v\rVert$$
$$\forall u,v\quad |\langle u,v\rangle|\leq \lVert u\rVert\lVert v\rVert$$
are equivalent. Indeed, the second implies the first, and consider successively $u$ and $-u$ in the first to get the second one.