What is the difference between an equation written in function notation and one that is not?

An equation is an equality which is satisfied by a unique set of values of your variables. You have, after the ##=## sign a fixed value, a fixed result.

For example: the equation ##4x-2=0## has zero as result and only ##x=1/2## as solution; this means that if you substitute the value of ##x=1/2## in the equation you have the result zero, i.e., the equation is satisfied.

Now, a function is similar, the only difference is that now you can have a lot of results after the ##=## sign and so you can have a lot of solutions.

For example: the function ##4x-2=y## doesn't have a definite result (as before that was zero) but another variable ##y##, so every time you choose an ##x## you'll get the corresponding value of ##y## that satisfies it. If you choose: ##x=1 -> y=2## ##x=2 -> y=6## ....etc.

If ##x=1/2 -> y=0## which is the solution that we found before for our specific equation (in which the ##y## was already set as zero)!

So to summarize, an equation has a fixed result (after the ##=## sign) and an unique set of solutions (values of the variables); a function can have a lot of results (possibly ##oo##) and, as a consequence, a lot of solutions.

hope it helps