In this answer I wrote that perhaps the principal utility of canonical forms is to tell whether two things are equal.
Is $\dfrac 1 {1-\sqrt 2}$ equal to $1+\sqrt2$, or $\dfrac 1 {2\sqrt 3}$ to $\dfrac{\sqrt3} 6$? Just put the expression in "simplest radical form" and you can see whether they're equal.
Is there another purpose?
Is the quadratic equation $\dfrac 5 x = 3+x$ the same as $x^2+3x-5=0$? Put them in the form of quadratic term first, then linear term, then constant term.
Is $\dfrac{51}{68}$ equal to $\dfrac{21}{28}$? Put them in lowest terms.
Perhaps another purpose can be seen in this: $$ \left( \frac{\sqrt 2\,\cos x -1 }{\cos x-\sqrt 2} \right)^2 + \left( \frac{\sin x}{\cos x - \sqrt 2}\right)^2 = 1 $$ is better than the form $$ \left( \frac{\sqrt 2\,\cos x -1 }{\cos x-\sqrt 2} \right)^2 + \left( \frac{\sin x}{\sqrt 2 - \cos x}\right)^2 = 1 $$ despite their saying the same thing.
Likewise $$ \text{area} = \frac 1 4 \sqrt{\vphantom{\frac 1 1} (a+b+c)(a+b-c)(b+c-a)(c+a-b)} $$ is better than $$ \text{area} = \frac 1 4 \sqrt{\vphantom{\frac 1 1} (a+b+c)(a+b-c)(b+c-a)(a+c-b)} $$ because of the rotation of $a,b,c.$ Does this count as a different purpose? Or might it be a purpose of a different thing?
$$\frac1{1-\sqrt2}=1+\sqrt2\iff \color{red}1=(1-\sqrt2)(1+\sqrt2)=\color{red}{-1}$$
Ok, so that was just goofing around a trivial typing mistake. To try to actually and seriously address your nice question:
No, the equation $\;\frac5x=3+x\;$ is not the same equation as $\;x^2+3x-5=0\;$...not even close: the first equation has a rational non-polynomial fucntion on the left side whereas the second expression above is a polynomial . What happens is that we usually are interested in the equations' solutions , and both equations above have the very same solutions, though one of them is defined on onje point more than the other one.
About your question about fractions: simple definition, and
$$\frac{51}{68}=\frac{21}{28}\iff51\cdot28=21\cdot68(=1,428)\;\;\;\color{green}\checkmark$$
Of course, you know all this and all the rest of things you "asked" (better wondered), and I'm not sure I can see what the actual intention of all this could be, yet when we have some uses we go with one thing, and for other ones we may go with another one
FOr example, if there are $\;68\;$ people and we bought $\;51\;$ pizzas to share among them all, it may be easier and much clearer to actually write $\;\frac{51}{68}\;$ instead of an apparentely easier $\;\frac34\;$. The last fraction tells me very little in the first exposed situation.
I insist with my students, either from univeristy or from hig school (when I have them), that if possible and reazonably easy and quick they must write an expression (and I'm thinking of functions now" in several equivalent ways depending on what the task is, for example
$$f(x)=x-\frac1x=\frac{x^2-1}x=\frac{(x-1)(x+1)}x$$
The first form is nice to realize what the domain is and, more important, to differentiate it in case of necessity. The second and third forms are better to find out where the function vanishese and also for asymptotes .
Finally, whether $\;c+a-b\;$ is better or worse than $\;a+c-b\;$ or $\;a-b+c\;$ is mostly a matter of tste, though I think that alphabetical order usually makes things easier to grasp, so I'd go with the third formk unless some conditions are given that may make other form easier to work with.