When I asked this question of some people with experience implementing proof assistants, their answer was "eta-laws".
If you test equality using only some kind of reduction algorithm, then generally speaking your equality will only incorporate beta-reductions such as $(\lambda x. M)(N) \equiv M[N/x]$. Eta-equivalences such as $M \equiv (\lambda x. M x)$ are quite difficult to implement with a reduction-only algorithm. There are various reasons for this, such as the fact that if you want to reduce $\lambda x. M x$ to $M$ you need to check that $x$ doesn't occur in $M$, and if you want to expand $M$ to $\lambda x. M x$ you need to have type information present to know that $M$ has a function-type. I won't say it's impossible, but normalization-by-evaluation is a clean, generalizable, and easy-to-reason-about family of algorithms that perform both $\beta$-reduction and $\eta$-expansion in a type-directed way.
To be sure, I believe this answer depends on a somewhat broader meaning of "normalization by evaluation" than is sometimes used. For instance, this property doesn't depend on the representation of values of function-type as actual metalanguage functions; it's sufficient to defunctionalize them and we can still call that "normalization by evaluation" for this purpose.
In addition, one doesn't have to check $\beta\eta$-equality by first normalizing to a $\beta\eta$-normal form and then doing an $\alpha$-equivalence check: one can incorporate the two $\beta$ and $\eta$ stages directly into a "bidirectional" equality-checking algorithm. But for purposes of this answer (which is, again, not one that I invented, but was given by those in the know) we also consider that to be a form of "normalization by evaluation".