A sum of $K$ unit fractions, each of denominator $n_i \leq n$, can
be rewritten as a fraction with a denominator bounded by
the product of the $n_i$, i.e. by $n^K$. (A small improvement is possible, with product of distinct integers $\leq n$)

A difference of two such fractions

(which are themselves sums of $K_1$ and $K_2$ fractions, respectively),
is a fraction with a demominator bounded by $n^{K_1+K_2}$.
(If $K_1\neq K_2$ the order of magnitude of the difference is actually
$\frac{1}{n^{\min(K_1,K_2)}}$.)

Let us assume that $K_1=K_2$.
So, the smallest nonzero difference $d(K,n)$ of sums of
$K$ unit fractions,
with $n_i \leq n$ is $d(K, n)\leq \frac{1}{n^{2K}}$.

I believe that this order of magnitude, with slightly weaker constants,
can be achieved with a constructive parametrization.

Example: If $K_1=K_2=2$, then choose
$\frac{1}{x^2 + 4 x + 1} + \frac{1}{x^2 + 4 x + 3} -
\frac{1}{x^2 + 3 x + 1} - \frac{1}{x^2 + 5 x + 5}=
\frac{2}{(1 + 3 x + x^2) (1 + 4 x + x^2) (3 + 4 x + x^2) (5 + 5 x + x^2)}$.
Now, taking $n=x^2+5x+5$, one has a set of 4 unit fractions,
for which the difference of the sums above is asymptotically
$ \frac{2}{n^4}$. This example is (for $K_1=K_2=2$)
possibly the best one can find, (but I did not prove this).

I conjecture that for larger values of
$K$ one can construct similar polynomial examples.

Quite possibly this has applications in questions in diophantine approximation,
exponential sums, large sieve etc.