Overruling is a finding about the law. Distinguishing is a finding about the facts. They are different operations, and treating one as a weaker version of the other is how good authorities get abandoned and dead ones get filed.
Overruled. A bench competent to do so has held that the earlier view was wrong in law. The earlier judgment no longer states the law on that point, and it does not matter how carefully it was reasoned or how often it was followed before. Overruling need not use the word. Where a larger bench lays down a proposition that cannot stand together with an earlier judgment, the earlier judgment is overruled by necessary implication, and nothing in the text will say so. This is the treatment most easily missed by a search that looks only for the word.
Distinguished. A later court has accepted the earlier judgment as correct and held that it does not govern the case before it, because the facts or the statutory setting are materially different. The authority survives untouched. What has changed is the map of where it applies. An authority distinguished thirty times is still good law and is also, in practice, close to unusable outside its original facts — a fact that the phrase "still good law" completely conceals.
The practical test is simple. Ask what the later court said about the proposition. If it said the proposition is wrong, that is overruling. If it said the proposition is right but does not reach these facts, that is distinguishing. If it said the proposition may be wrong and someone larger should decide, that is a reference — and the honest answer to your client is that the point is open.
A third case sits between them and is worth naming: a judgment that has been explained. A later court restates what the earlier judgment meant, often narrowing it without saying it was wrong. Explanation is not overruling, but a proposition that has been explained twice is rarely still as wide as its headnote.