Why is the definition of species important and what is the use for the definition of species?
The definition of species is important for a few reasons. 1. It is needed for measuring diversity of a community/ecosystem (currently) 2. With many species, it's not a completely arbitrary distinction (mostly sexual species). There are some fundamental biological differences between groups of organisms that are defined as species, and understanding the biological processes that create these distinctions is useful 3. For practical reasons: it's important in the literature that when I say species I mean roughly the same thing as when you say species. Otherwise, we're constantly have to describe in detail exactly what we mean
However, as you've already brought up, defining species is much easier when it comes to sexual organisms that asexual ones. Even if hybrids exist, defining the species is usually not that difficult. The hard part comes with things like bacteria, when it become very arbitrary. Field microbiologists and ecologist often don't use species, they use operational taxonomic units (OTUs) which are defined by an arbitrary cut off of between 80-95% similarity of the 16S rRNA gene. The question really becomes, is species a useful concept for all organisms? I personally think that it is for some, and perhaps not for others. It may be that for asexual organisms, the idea of species is actually not useful, but that doesn't mean we can't use the concept for sexual species.
We have many intuitions in our life and the point is that many of these intuitions are wrong. The question is, are we going to test those intuitions? -Dan Ariely