For a positive integer $n$, let $P(n)$ be the set of distinct prime divisors of $n$. We are looking for pairs of numbers $A,B$ such that $P(A)=P(B)$ and $P(A+1)=P(B+1)$. The pair $A=2$, $B=8$ is a simple example.
There are infinitely many such pairs given by $A=2^m-2$ and $B=2^mA$, for any integer $m>1$. (this is how I solved a problem asking whether there exist infinitely many such pairs. I'm pretty sure the problem was in one of the Miklós Schweitzer competitions, but I can't for the life of me find it now).
Another "sporadic" solution is given by $A=75=3\times 5^2, B=1215=3^5\times 5$. Miraculously, $A+1=2^2\times 19$ and $B+1 = 2^6 \times 19$. I'm unable to fit this coincidence into an infinite family of solutions.
Questions: Are there any additional pairs? Can we prove that there aren't any? Can we at least prove that there are only finitely many such pairs, except for the infinite family I mentioned?