On the other hand, they were not just different populations of the same species, like bulldogs and spaniels.
There is no perfect definition of a “species”. There is a taxonomical model that groups Homo sapiens into different *subspecies* — so modern humans are Homo sapiens *sapiens*, Neanderthals were Homo sapiens neanderthalensis. It’s not a question of which taxonomical definition is correct — that is subjective. It would not be necessarily wrong to class sapiens and Neanderthals as the same species, just as it is not necessarily wrong to group them as separate species.