Your title vs. explanation differ.
But, if you have the same structure you can use both methods to find missing data.
EXCEPT
is a specific implementation that enforces same structure and is a subtract operation, whereas
LEFT ANTI JOIN
allows different structures as you would say, but can give the same result.
Use cases differ: 1) Left Anti Join can apply to many situations pertaining to missing data - customers with no orders (yet), orphans in a database. 2) Except is for subtracting things, e.g. Machine Learning splitting data into test- and training sets.
Performance should not be a real deal breaker as they are different use cases in general and therefore difficult to compare. Except will involve the same data source whereas LAJ will involve different data sources.