First off, grassroots vs astroturf describes a spectrum, rather than a binary or dichotomy.
Astroturf implies control and/or financing by an organization. That organization might pay participants (Brooks Brothers Riot) or pay for advertising to generate support from those who would otherwise be apathetic.
It is said that the Tea Party started out as grassroots, but was coopted and turned into an astroturf operation. OTOH, Occupy Wall Street rejected any kind of centralization and can thus be described as almost purely grassroots.
Astroturfers often have additional, hidden goals that may be misaligned with those of the supporters that they have attracted.