What is the difference between astroturf and artificial grass?

The short answer to your question: AstroTurf was artificial grass! 

AstroTurf is more of ancient name, it is known because of astrodome. Astrodome was built in 1965, the first indoor multipurpose sport facility! Once the dome was built, real grass was installed, but died. The carpet industry made the artificial turf to replace the real grass inside the astrodome known as “AstroTurf”. It was when artificial grass was born and nobody would call the product “artificial grass”, but refer it as “AstroTurf”, the name become an alternative of the product itself ever since. However, during the past 55 years, artificial grass has gradually become an industry and more and more people start referring to the “artificial grass” or “synthetic turf” than using “AstroTurf”. This is more like personal computers: IBM used to be the alternative name for pc when the company invented it, but after years of industry growing, people now refer “pc” more than IBM or at all. Therefore, AstroTurf is more of the old name, not a trending or current alternative name for artificial grass. People in the sport world seems to be more familiar with AstroTurf because of astrodome, and today “older” generations might still refer artificial turf used at sport fields as AstroTurf, but that is not necessarily the case for landscape artificial grasses. The artificial grass industry has completed reinvented itself: today’s artificial grass simulates almost perfect as real grass. AstroTurf was more of plastic carpet with very short piles and not even any close to be called grass. But being the pioneer and well known, some people still use it as an substituting name for artificial turf especially in the sports field world.

Astro is a masculine noun; it means "star" in astronomy or "aster" in botany. Most likely, the original idea was associated with celestial bodies, Astro in Greek, aster in Latin. A combining form must have meaning "about stars or celestial bodies, or activities, taking place outside the earth's atmosphere" - astrology, astronaut, astrophysics, astronomy, dis"aster" :-) But not in the case of astroturf, or what has derived from it. 

The "astroturf" came from "astrodome" in the 1960s. In 1993, it originated the word "astroturfing" - the practice of using "fake grassroots" to influence public opinion. As wiki said, "Astroturf-ing" is the 21st century's version of false advertising. Astroturf made the word "Astro" equal to "fake," at least in combination with "turf." If we apply the same rules to all "Astro"-words, astro-logy becomes a fake-logy; -logy in Latin means "speaking, treatise, science, theory." Astro-nomy - fake-rules (-nomy means a system of rules, laws, or knowledge). How about an astronaut? We are replacing a "star" with "fake," don't we? It's one of those neologisms in English vocabulary that don't make sense. Like strawberry, jackpot, or wedlock. 

AstroTurf is a brand name, artificial grass is the product. Since the artificial turf industry started with the turf used in astrodome, people have been using AstroTurf as an alternative name for artificial grass for a long while. However, AstroTurf has gone through a wild ride during the past 50 years. The company behind the brands had filed bankruptcy, and the ownership of this brand have been changed many times as sports business world still views this brand with value and trying to continue its legacy. Now AstroTurf brand is owned by a German company called Sports Group. Since the artificial grass industry has emerging to a new level now days, the image of old AstroTurf with short pile might still buried in the generations who have lived through its time, but artificial grass should not equal to AstroTurf any more in my opinion: today’s artificial grass comparing with original AstroTurf turf is really day and night, and new generations including some industry professionals might not be familiar with AstroTurf any more, including people like you who posted this question. 

AstroTurf became a famous brand because of astrodome, its legacy is becoming a past tense along with the fate of astrodome which was closed in early 2000’s and became a historical site: No more astrodome’s glory, so starts the fading of AstroTurf. Even generation who are familiar with AstroTurf nowadays will mention AstroTurf only to emphasize how today’s artificial grass is so different from the “old” AstroTurf which was literally plastic carpet. 

In my opinion, the glory of AstroTurf has started fading away regardless the efforts of its current ownership trying to prolong this brand’s lifespan. The bankruptcies (companies behind this brand had filed at least two bankruptcies in the past) along with the closure of astrodome have made it difficult to maintain and further develop this old brand, especially the current industry’s mainstream has been more focused on products that replicate real grasses and trying to differentiate their products from the old image of AstroTurf, such drive will further separate artificial grass itself from the ancient name of AstroTurf going forward.