Typically, species are defined in terms of a population's inability to produce viable offspring with a different population. How, then, do we define the species with regard to asexual creatures? Is it gross genetic difference? Phenotypical difference?

Yeah, they're different species when we consider them different enough (morphological species). Same for other species that don't reproduce sexually (anymore), such as extinct species. The definition also works for species that do reproduce sexually, but won't always match the biological definition.

Well, that is one of the main problems of species definitions. In prokaryotes sequence identity of 70% (IIRC) as detected by DNA-DNA hybridization was considered to be a gold standard to distinguish species. Nowadays this is corroborated using molecular biological markers. The morphological species definition is highly problematic and usually only works somewhat when certain traits are used that have been well characterized in a phylogenetic context. For most asexually reproducing organisms this is most likely not the case.