The source of the confusion is that you are not dealing merely with implications in propositional logic, where, as you said, the truth tables ensure that $(p\implies q)\lor(q\implies p)$ is always true. Your statements about angles are universally quantified statements, even though the English language lets you hide the quantifiers. Your first statement really means "For every two angles $x$ and $y$, if $x$ and $y$ are congruent then $x$ and $y$ are not equal." Similarly for the second statement.
So the logical form of these statements is $(\forall x)(\forall y)\,(P(x,y)\implies Q(x,y))$ and $(\forall x)(\forall y)\,(Q(x,y)\implies P(x,y))$. Because of the quantifiers, it is entirely possible for both of these to be false. All that's needed for that to happen is that some particular $x_0$ and $y_0$ satisfy $P$ but not $Q$, while a different pair $x_1$ and $y_1$ satisfy $Q$ but not $P$.
If quantifiers are not involved, for example if you have two particular angles and are making statements about just this pair, not angles in general, then a proposition and its converse will not both be false.