The molecules which make up those neurons come and go on a regular basis. The patterns that the molecules are set in remains relatively the same.
And it is the pattern, not the molecules or atoms, that are the person.
I am not sure that it is even sensible to discuss whether one hydrogen atom is 'the same' as another; If you swap two hydrogen atoms, all else being equal, then the system is identical to how it would be if you didn't; and due to quantum uncertainty, two hydrogens that are close to each other may have a significant overlap in their probable positions.
As anyone whose old broom has had three new heads and two new handles can tell you, the identity of a whole is more complex than the sum of the identities of its parts. That is even more true of dynamic systems. An ocean wave is clearly an entitiy in its own right; one can surf it for hundreds of metres, and any surfer will tell you he has stayed on the same wave all that time; You can even track it across thousands of km of open ocean. But the water molecules in the wave don't move much at all - the ones that started out 100 metres from the beach mostly stay about 100 metres from the beach, and the ones that are part of the wave as it breaks on the shore mostly stay within a few metres of the shore.
Life is both dynamic and (to some degree) self-repairing. A broom that replaces its own worn out handles, if you like. This is only surprising to humans because we live at a scale where identity is ubiquitous - Even two pool balls are never identical, as the pattern of scratches on their surfaces are unique. We are simply not equipped by our experience to truly grasp that identity of particles is ultimately meaningless - that you can't put a chalk mark on a proton, feed it into a living cell, and come back later to see where that particular proton ended up. In a very real sense, there is no such thing as 'that' proton. There are just protons.