Yup, at least according to the article:
"His paper shows that there is some number N smaller than 70 million such that there are infinitely many pairs of primes that differ by N. [...] [N]o matter how sparse the primes become — you will keep finding prime pairs that differ by less than 70 million."
So he proved that it there is some number N, not necessarily 70mil but below 70mil, is the "gap" between primes.
This is an amazing development. On the other hand, we are also getting close to nailing the odd Goldbach Conjecture. I think it was just last year that Tao proved, without the Riemann Hypothesis, that every odd number N > 1 can be expressed at by the sum at most 5 primes. Wonderful to witness such great leaps in maths during one's own lifetime.
He showed that there are infinitely many twin primes differing by 70 million from each. However, it could be that the next such twin prime, has its lower prime number more than 70million numbers further along.
What I mean is, there is allowed to be a gap larger than 70million with no primes in it, then all of a sudden, twin primes. But those twin primes with a gap less than 70million between them are guaranteed.
Basically, if you separate all adjacent pairs of primes, with no primes in between, you can separate this into two groups of those adjacent pairs with a gap less or equal to 70mil, or greater than 70mil. The first group has infinitely many members according to the proof. And the second group is probably not empty.
We know for sure it's non-empty by simple factorial arguments ITT or by appeal to the prime number theorem.