Except that turned out not to be quite true, didn't it? The boundaries of what exactly one might want to consider a "planet" turned out to be fuzzy, and had to be set arbitrarily.
The basic concept of species is pretty straightforward, at least for sexually reproducing organisms. A species is a population that evolves together as a group, exchanging DNA each generation, so that traits can be passed around the whole population. One species becomes two when a population like that splits into two and each starts on an independent evolutionary trajectory. Deciding exactly when they have split enough that you should call them different species is pretty arbitrary -- what if there is only occasional interbreeding between the groups, for example?
When there's just one population and that species changes a lot over time, it's even more arbitrary exactly when it's changed enough that you call it a new species, since there's no interbreeding test to make.
The fundamental reason why species is difficult to define, then, is that it is applying a yes/no categorization to an inherently gradual continuum.