What is the proper way to verify that two expressions are equal?

You are using SameQ which does a direct structural comparison rather than a mathematical one. Since the expressions are not exactly the same it returns False. Try Equal:

FullSimplify[Abs[1/x + x^2] == Abs[1 + x^3]/Abs[x]]

True

FullSimplify is needed for nontrivial comparisons; without it Mathematica will return the equality as given if it is not trivially equivalent.

You can also use ForAll and Resolve which I think is sometimes faster (no example at hand):

ForAll[x, x != 0, Abs[1/x + x^2] == Abs[1 + x^3]/Abs[x]] // Resolve

True

Answer

Given two symbolic expressions a and b in Mathematica, the way to check equivalence is:

True === FullSimplify[a == b]

Explanation

The FullSimplify function is a thorough symbolic restructuring command that will be most likely to reduce two equivalent symbolic expressions to True.

The === operator will call SameQ, which will return True if both operands are exactly equivalent without any manipulation and False otherwise.

The reason for adding the True === can be seen with an example.

a = d/2;
b = 4c;
FullSimplify[a == b]

 8c == d

That's not what we want though, and if we put this output in an If statement it will not work correctly. Since we want the only possible outputs to be True or False, we write:

a = d/2;
b = 4c;
True === FullSimplify[a == b]

 False

Yay!