Space is significant

Empty space is often significant in mathematical notation, but it's easy to misinterpret, and hard to produce accurately.

TeX applies different amounts of spacing around symbols based on context. The ubiquity of TeX means that typeset expressions without this spacing look odd.

There's some evidence that people learning algebra use the spacing of symbols to deduce rules.

Producing reliable spacing in handwriting is hard.

Examples
Entries in a matrix are separated by empty space :

\[ \begin{pmatrix} 1 & - 2 \\ 3 & - 4 \end{pmatrix} \]

\[ \begin{pmatrix} 1-2 \\ 3 - 4 \end{pmatrix} \]

Statements about a line are usually separated from the main expression by a large space.

\[ 4 \equiv 1 \mod 3 \]

versus

\[ 4 \equiv 1 \bmod 3\]

Juxtaposing two fractions:

\[ \frac{1}{2} \frac{-3}{4} \]

\[ \frac{1-3}{2\, 4} \]