I don't agree. I think they are fairly interchangeable for the most part. I am quite sure that the English "all" translates to "alle" in Danish more often than "alt".
Jeg har spist [alt / alting]
Der findes [alt / alting] i den have.
Fortæl mig [alt / alting]
[Alt / alting] går galt når man hedder Kurt til fornavn.
The meaning is pretty much the same whether you use alt or alting. "Alting" carries a stronger meaning of "every. single. damn. thing" sometimes, but that is about it. I would use "everything" in all of the sentences above no matter if the Danish version contains alt or alting. Of course you can say "I have eaten it all", but that translates to "jeg har spist det hele"". To me, this is how it works:
Everything = Alt / Alting
All Things = Alle ting
Anything = Hvad som helst / Noget som helst
Nothing = Ingenting / Intet
All = Alle
All of it = Det hele