Term that can distinguish beta-equivalent normal forms in the untyped lambda calculus
Let's split it into two parts:

First let $m$ and $n$ be terms that have a normal form (but are not necessarily in normal form). Let
\begin{align*}
 \mathbf{A} &amp;= \{x \;|\; x =_\beta m\} \\
 \mathbf{B} &amp;= \{x \;|\; x \neq_\beta m \mbox{ and $x$ has a normal form }\}
\end{align*}
These sets are nonempty, disjoint and closed under $\beta$-equivalence, so 
we can apply Scott-Curry theorem and conclude that they're inseparable. So there is no algorithm that can determine if two such terms $m$ and $n$ are equal or not.

For the second part: Intuitively, a lambda term cannot distinguish if its arguments are in normal form or not. Suppose we have $f$ that can determine if two terms in normal form $m$ and $n$ are equal or not. Now let $m'$ and $n'$ be terms that have a normal form. We can use a normalizing reduction strategy to reduce them to their normal forms (which is accomplished in a finite number of steps), apply $f$ and determine if they're equal or not, which contradict the previous part.