TL;DR: I would say the difference is the level of confidence in the results of type-checking you'll get.

Of course an external tool could model and check against a proper type system, in theory. So that's indeed not the point.

A proper static type system comes with a formal definition and (math-like) proves of soundness (and progress). This will give you the guaranty that a well typed program is sound.

A "linter" can't give you that guaranty. All it can do is to say "LGTM"; but your program could still crash or bug-out in some other way at runtime. That's quite a big difference imho.

Linters are more like automated tests: They may tell you something is wrong, but they don't give any guaranties that things are actually right. A proper static type system OTOH gives such guaranties.

For example I would consider TypeScript being only a very funky linter—as it does not model a sound type system. My understanding is that mypy and friends are not different. All those linters produce not only way to much false positives, no, they also produce false negatives (meaning that they let unsound programs pass; something that can not happen with a formally verified type system).

As an anecdote, I got bitten by that when trying TS (and it was a very sad experience): When playing around with TS for the first time I did not know that TypeScript is barely a linter. I was really very excited about all those things you can do with proper structural types there! I was coming from Scala 2 which has only very limited support (based on JVM reflection) for this kind of types¹. But I always loved the "objects as hash-maps" concept of JS… So I've written some small REST client for one of our back-end services in TS. Everything nicely typed, no escape hatches used. Someone also told me I have to use "stric mode" which will "catch all type errors". That looked a bit odd, but OK JS was not made to be typed in the first place. Everything looked great than. As someone coming form the purely functional Scala camp (where things work when they compile :-D), and encouraged by all those people praising TS for its "powerful type system", I've put the same amount of confidence as in Scala into the typed code I've written in TS. By this I mean you don't need to double check whether things are correct if the typer says they're correct. It's enough to check the logic of your program. You don't have to try out "manually" (this includes written tests!) every line of code. Up to than everything looked good. So I've pushed things and started to do end-to-end test. That was when I fell out of my dress. My code produced runtime errors which where actually "impossible"! The TS "type-checker" said things are such and such but at runtime they were different (which I found out only after some deep debugging sessions as this was something I did not expect in any way)! I think I've wasted one and a half day debugging this shit only to find out that TS "types" are actually worthless. If I have to try out any and every line of code manual anyway what's the point of TS? I could have written this thing in pure JS (where I have to also check any and every line manually), but it would have been much easier without "wrestling" with that quite useless "type system"… :-(

The point is: A dynamic language remains a dynamic language even if you use some linter! Without true guaranties you can't have much additional confidence in your code without testing just everything—even the parts that were already "tested" by our "type system". Actually it's even worse than without a linter if you don't know about its shortcomings and make the mistake to blindly trust such a tool; like you can do in case of a sound type system (modulo the very seldom cases of compiler bugs).

I hope I could make it now more understandable what I mean by "proper static type system".

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¹ Things improved in Scala 3 though.