Index notation for tensors: is the spacing important?

It's important to keep track of the ordering if you want to use a metric to raise and lower indices freely (without explicitly writing out $g_{ij}$'s all the time).

For example (using Penrose abstract index notation), if you raise the index $a$ on the tensor $K_{ab}$, then you get $K^a{}_b (=g^{ac} K_{cb})$, whereas if you raise the index $a$ on the tensor $K_{ba}$, you get $K_b{}^a (=g^{ac}K_{bc})$. Since the tensors $K^a{}_b$ and $K_b{}^a$ act differently on $X_a Y^b$ (unless $K$ happens to be symmetric, i.e., $K_{ab}=K_{ba}$), one doesn't want to denote them both by $K^a_b$.

Tensors can be thought of as multi linear maps from copies of a vector space (and its dual) to a field (usually $\mathbb C$). The placements of the indices tell you which "argument" goes where. E.g. $A_{mn} u^m v^n$ is not the same as $A_{nm}u^m v^v$. Perhaps Penrose's pictorial notation makes this clearest.