\begin{align}
4635^2 + 4292^2 &= 6317^2 & 4635 - 4292 &= 343 = 7^3 \\
3445^2 + 708^2 &= 3517^2 & 3445 - 708 &= 2737 = 7\cdot 17 \cdot 23\\
31005^2 + 28268^2 &= 41957^2 & 31005 - 28268 &= 2737 = 7\cdot 17 \cdot 23
\end{align}
The difference $a-b$ of the legs in a primitive Pythagorean triple can have arbitrarily many prime factors (distinct or not). But to have many prime factors the difference must be large, and hence the legs must be large too, so one first encounters many cases with few prime factors.
How to find triples such that the difference has many prime factors:
Recall that one can parametrise the primitive Pythagorean triples by
$$(r^2-s^2, 2rs, r^2+s^2)$$
with $r$ and $s$ coprime and of different parity. Thus the difference "odd leg $-$ even leg" is
$$(r^2 - s^2) - 2rs = (r - s)^2 - 2s^2\,.$$
If $r$ and $s$ are coprime, then $r-s$ and $s$ are coprime too, and a prime $p$ can divide the difference of the legs of a primitive Pythagorean triple if and only if $2$ is a square modulo $p$, that is the case if and only if $p \equiv \pm 1 \pmod{8}$. Every such prime can be written in the form $x^2 - 2y^2$ with $x,y$ (positive) integers, e.g. $7 = 3^2 - 2\cdot 1^2$, $17 = 5^2 - 2\cdot 2^2$, $23 = 5^2 - 2\cdot 1^2$.
We can factor such a representation as $p = (x + y\sqrt{2})\cdot (x - y\sqrt{2})$, and the set
$$\mathbb{Z}[\sqrt{2}] = \{ x + y\sqrt{2} : x,y \in \mathbb{Z}\}$$
is a ring, in particular the product of two such numbers is again of that form: $$(x+y\sqrt{2})(u + v\sqrt{2}) = (xu + 2yv) + (xv + yu)\sqrt{2}\,.$$
And $(x-y\sqrt{2})(u-v\sqrt{2}) = (xu + 2yv) - (xv + yu)\sqrt{2}$, so
$$(x^2 - 2y^2)(u^2 - 2v^2) = (xu + 2yv)^2 - 2(xv + yu)^2\,.$$
Thus if you have distinct primes $p_1, \dotsc, p_r$ with representations $p_{\rho} = x_{\rho}^2 - 2y_{\rho}^2$, then
$$x + y\sqrt{2} = \prod_{\rho = 1}^r (x_{\rho} + y_{\rho}\sqrt{2})$$
gives us a pair of positive integers with
$$x^2 - 2y^2 = \prod_{\rho = 1}^r (x_{\rho}^2 - 2y_{\rho}^2) = \prod_{\rho = 1}^r p_{\rho}\,.$$
Since the $p_{\rho}$ are by assumption all distinct, $x$ and $y$ are coprime - if not all $p_{\rho}$ are distinct, one must choose the signs of $x_{\rho}$ and $y_{\rho}$ corresponding to the same prime in a compatible way to have $\gcd(x,y) = 1$, but that can always be done - and $x$ is odd. Then $r = \lvert x \rvert +\lvert y\rvert$ and $s = \lvert y\rvert$ produces a primitive Pythagorean triple such that the difference between the legs is $p_1\cdot \dotsc \cdot p_r$.
As the first example at the top shows, it is not necessary that all the primes are distinct, we can find primitive Pythagorean triples for which the difference between the legs is divisble by a power of some prime, and we can have the exponent as large as we please. The second and third example have the smallest possible difference between the legs such that the difference is divisible be three distinct primes. The second corresponds to $(3+\sqrt{2})(5+\sqrt{2})(5-2\sqrt{2})$ and the third to $(3+\sqrt{2})(5+\sqrt{2})(5+2\sqrt{2})$.
