I'm not sure what a definitive answer would look like, but maybe I can frame the question a little differently in a series of comments:

1. The underlying question really concerns an arbitrary pair of (not too small) simple groups $G, H$ for which $\pi(G) = \pi(H)$. As observed by others, the classification of finite (nonabelian) simple groups should be in hand already. Given only finitely many groups non-isomorphic to an alternating group or group of Lie type, the question basically involves $G, H$ of those two types. (But there is no obvious reason to require that one of the groups be alternating.)

2. There is a long tradition, going back at least to Emil Artin in the 1950s, of sorting out which of the known finite simple groups having different origins are actually isomorphic. So the question asked is not really about existence of isomorphisms: one knows already from the literature which groups of Lie type are isomorphic to others of Lie type or to alternating groups. In practice all such accidental isomorphisms involve relatively small orders.

3. It's conceivable that some isolated group of Lie type will turn up whose order accidentally has the same prime divisors as some (not too small) alternating group. This would be a bit surprising, but probably not interesting unless something systematic is observed that involves more groups. The key word here is accidental. On the other hand, to prove rigorously that no accidents happen looks challenging, even if ultimately not too illuminating.

4. Why emphasize alternating groups? There is perhaps one good reason: here the primes dividing the group order form a list (from 2 to the largest prime) having no gaps. For groups of Lie type that are not very small, this behavior of $\pi(G)$ strikes me as purely accidental and somewhat unusual. If that's not the case, it would be interesting and need an explanation. But considering a family of groups such as $\rm{PSL}(2,p)$ (with $p>3$) suggests a skeptical attitude.

I'm not sure what a definitive answer would look like, but maybe I can frame the question a little differently in a series of comments:

1. The underlying question really concerns an arbitrary pair of (not too small) simple groups $G, H$ for which $\pi(G) = \pi(H)$. As observed by others, the classification of finite (nonabelian) simple groups should be in hand already. Given only finitely many groups non-isomorphic to an alternating group or group of Lie type, the question basically involves $G, H$ of those two types. (But there is no obvious reason to require that one of the groups be alternating.)

2. There is a long tradition, going back at least to Emil Artin in the 1950s, of sorting out which of the known finite simple groups having different origins are actually isomorphic. So the question asked is not really about existence of isomorphisms: one knows already from the literature which groups of Lie type are isomorphic to others of Lie type or to alternating groups. In practice all such accidental isomorphisms involve relatively small orders. (On the other hand, simple groups coming from odd orthogonal groups and symplectic groups with the same rank and same finite field are well understood to be non-isomorphic in spite of having the same order.)

3. It's conceivable that some isolated group of Lie type will turn up whose order accidentally has the same prime divisors as some (not too small) alternating group. This would be a bit surprising, but probably not interesting unless something systematic is observed that involves more groups. The key word here is accidental. To prove rigorously that no accidents happen looks challenging, even if ultimately not too illuminating.

4. Why emphasize alternating groups? There is perhaps one good reason: here the primes dividing the group order form a list (from 2 to the largest prime) having no gaps. For groups of Lie type that are not very small, this behavior of $\pi(G)$ strikes me as purely accidental and somewhat unusual. If that's not the case, it would be interesting and need an explanation. But considering a family of groups such as $\rm{PSL}(2,p)$ (with $p>3$) suggests a skeptical attitude.