One of the threads amongst more radical politicos in early post-Revolutionary America was that the Common law itself was in some ways unjust. Not the system itself (judges presiding over cases and making decisions which are, to some extent, binding on future judges, and which have the same effect as statutory law), but the body of decisions rooted in English common law. It was thought that the entire point of the Revolution was to get rid of English jurisprudence in America.
So, what would have changed had the US started over with its own common law? Developing its own body of law based on decisions made by American judges, that is.
There are sure to be tons of changes, but the only one that immediately comes to mind is related to the situation of labor. In this time period, labor organizations (like unions) could be criminally prosecuted under a common law doctrine that held labor unions to be a conspiracy to restrain trade (or something very similar). If various states didn't pass statutes assuming English common law, what happens here will depend on how the precedent starts getting set. It could go one of two ways:
1. In the areas where most judging is going to happen (the commercial areas in the north), the Federalists had a lot of control early on and they're going to be pushing a pro-business, anti-labor agenda of their own. We could end up seeing little difference between the English common law and the American one on this subject.
2. Alternatively, things could evolve in the other direction because, in early America, there was a cultural proclivity to celebrate the ordinary labor. Remember, this is before the massive immigration waves that introduced America to cheap labor, in a time when the price of labor was actually really high, and each individual laborer was a lot more valued by society. This is one of the times (Jeffersonian and then Jacksonian democracy) when the ordinary man was most politically powerful. It might be difficult to overcome the Federalist strangehold on city politics but once it's done, things might turn out rather well for workers.
What else might change?