  For a pair of distinct non-CM newforms of weights at least 2, having rational
integral Fourier coefficients $a_{1}(n)$ and $a_{2}(n)$, under GRH, we obtain
an estimate for the set of primes $p$ such that $$ \omega(a_1(p)-a_2(p)) \le [
7k+{1}/{2}+k^{1/5}],$$ where $\omega(n)$ denotes the number of distinct prime
divisors of an integer $n$ and $k$ is the maximum of their weights. As an
application of our result, under GRH, we show that the number of congruence
primes for any such newform is bounded by $[7k+{1}/{2}+k^{1/5}]$. We also
obtain a multiplicity one result for newforms via congruences. Finally, the
methods of this paper allow us to obtain the normal order for the function
$\omega(a_1(p)- a_2(p))$.
