If an opponent targets one of my creatures with a destroy spell, and in response I tap Nephalia Smuggler to exile that creature and return it to the battlefield, does the destroy spell fizzle due to its target having vanished? Or is the target considered still valid, since it's still on the battlefield after the effect of Nephalia Smuggler has been resolved?
If the spell only has one target, it will be countered on resolution since the creature it was targeting no longer exists.
First, a correction. The creature isn't still on the battlefield after the effect of Nephalia Smuggler has been resolved. The creature that's there then is a brand new creature with no relation to any creature that was previously on the battlefield. That the old and the new creature were created by the same card is irrelevant. [CR 400.7]
The answer to your question depends on the exact wording of the "destroy spell". What happens depends on if the spell targets, and on if it still has legal targets as it would resolve. If it has targets and all of them have become illegal, the spell is countered. Otherwise, the spell continues to resolve, doing as much as it can with its remaining targets. [CR 608.2b]
If the only target of the spell was the creature that Nephalia Smuggler exiled, then the spell would be countered on resolution. (Note: fizzle is now depreciated, and is no longer used within the context of the rules. It used to appear in older versions of the rules for historical reasons, but only in the Glossary.)
The creature that is on the battlefield after Nephalia Smuggler's ability resolves is a new Object. The spell that was targeting the old creature is not targeting this "new" one.
There are two possibilities. If the opponent's spell had multiple targets, it would resolve but would be unable to affect the creature that was exiled by Nephalia Smuggler (it would still have any other effects on the card). If the opponent's spell only had a single target, the creature that was exiled, then the spell would be countered on resolution and would have no effect. From the Basic Rulebook (page 10):
The creature exiled by Nephalia Smuggler left the battlefield and went to a different zone. When it did this, it became a new object, and hence this "new" creature it isn't being targeted by your opponents spell.