First of all, I would like to state that the English Premier League (EPL) is certainly of a higher standard, overall, than Major League Soccer (MLS). However, the gap is not as wide as many would believe. The evidence presented below shows that, for the neutral, an MLS game is often played at just as high of a level, from a quality of possession point of view, as an EPL game. MLS games have slightly more turnovers and slightly less attempts on goal and goals per game. But, if you exclude the first half of the year, the league characteristics are almost indistinguishable.
For me, measuring the quality of a team is simple: what do they do with their possessions? The best teams in the world make their possessions count. They attempt more shots per possession than lesser teams and they score more goals, all while limiting their opponent's shots per possession and goals. Tables for the EPL and MLS seasons bear this out. The best teams have the most shots per possession and give up the least shots per possession.
I would love to include more leagues or seasons in this comparison, but for now all I have to work with are the 2011-2012 EPL season and the 2012 MLS season. The most interesting feature from this analysis is the sudden and strange increase in quality from MLS in the middle of the season. This phenomenon was explored more here. The EPL did not experience such fluctuation in style of play. Weather might be a suspect, but it's not as if England doesn't experience changing seasons too.
Statistics presented are per team (i.e. 112.22 turnovers per game in EPL is per team, not match total).