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

Hint: Here, first we need to understand the equation that is written in a function notation like $y=f\left( x \right)$. We will consider some examples of equations like linear equations, quadratic equations. Now, we will see the equation that is not written in function notation by substituting $y=0$ in above taken examples of the assumed equations.

Complete step-by-step solution:
Here we have been asked to describe the difference between an equation that we write using a function notation and an equation that we do not write using a function notation. First we need to understand the meaning of the term ‘function’.
In mathematics, a function is a relation where for each value of the domain (set of values of x) there must not be more than one value of the range (set of values of y). It is written in the form $y=f\left( x \right)$ and is read as ‘y as a function of x’. Here for each value of x there is not more than one value of y. For example: - $y=4x+9$ is the notation of a linear equation, $y={{x}^{2}}+4x+8$ is the notation of a quadratic equation. These are some examples of the functions written in function notation.
Now, if we substitute the above functions equal to 0 to convert them in the form $f\left( x \right)=0$ then they will be called as functions written without the function notation. For example: - $4x+9=0$, ${{x}^{2}}+4x+8=0$.
You can now detect the main difference between the two notations. In case of $y=f\left( x \right)$ there will be many values of y as we can substitute different values of x to get different values of y. However, in case of $f\left( x \right)=0$ there will be some fixed values of x because the value of y has been fixed as 0. Like for the linear equation $4x+9=0$ there will be only one value of x while for $y=4x+9$ there will be infinite pairs of (x, y).

Note: Note that it even if you do not substitute $y=0$ and instead of that you substitute any other fixed value, in such case also we can convert them into the form $f\left( x \right)=0$ by taking all the terms to the L.H.S and making R.H.S equal to 0. Remember the definition of a function and remember that if for one value of x there are more than one values of y then that relation is not called as a function.