Are there practical applications to the new prime pair proof? I've recently heard that its been proven that the set of prime pairs that are separated by no more than 70,000,000 is infinite.
Does this have any impact on cryptography or another practical application?
 A: The twin prime conjecture (and the weaker form of Zhang's theorem) doesn't have (known) connections to any problem in cryptography. So the answer is no, for now. Not every problem connected to prime numbers have necessary an implication to cryptography.
At the moment (like other problems in number theory, for example the Goldbach's conjecture) there is no important practical applications in other sector of mathematics or other disciplines. But nothing prohibit that a day someone find an useful application.
A: What you say you recently heard is false. [I see you've edited the question in light of my answer, so now it says something that is actually true.] There are indeed pairs of primes separated by more than 70 million composite numbers.  What was proved is that there are infinitely many pairs of primes separated by no more than 70 million composite numbers.  The fact that it's true in infinitely many cases does not mean it's true in all cases.  For example, it is conjectured that there are infinitely many twin primes, i.e. primes separated by exactly $2$, so just one composite number is between them.  But if so, that obviously doesn't mean that EVERY pair of consecutive primes is separated by only two.
