They're both perfectly valid and "natural", and in most cases they'd be equivalent and interchangeable.
But potentially there could be a difference. If it turned out the Chinese consulate was closed when he got there (or there was some other reason why he couldn't apply for a visa), the second alternative (using the infinitive to apply) would still be valid. But in that scenario, the first version (with the and conjunction followed by a "tensed" verb form) would be incorrect (because he didn't actually make the application).
That's because in version #2, to apply [for a visa] is an adverbial clause defining purpose1 (the reason he went to the consulate) - he could still have gone there for that purpose even if he was unsuccessful in his intentions. But the and version unambiguously forces the interpretation that he was successful.
1a: Then he went to the consulate and applied for a visa, but it was closed. - WRONG!
2a: Then he went to the consulate to apply for a visa, but it was closed. - FINE
1 Noting the later question past simple + to and infinitive form, it's worth explicitly pointing out that to in such contexts is effectively "short for" in order to (hence it's called an "adverb of purpose").