So, what are the differences?
Are you serious?
Yes, I am.
Anarchy = no rules about anything, you do anything, random warriors of the night assaulting random people.
As I suspected, you have no clue what you are talking about: which definition are you using, defined by who?
This is the definition that the general media and public education gives you since you are a child, so that you can freely associate anarchy with chaos, killing, fraud and whatever.
The modern definition of anarchy upon the vast majority of "modern" (i.e.: of the last decades) anarchist agree upon is the one defined by Murray Rothbard in his writings, generally regarded as "anarco capitalism".
Of course anarchy has nothing to do with what you state, anarchy does not
means absence of rules, but absence of a central
power that can enforce arbitrary rules to the society. A part for a very small minority of anarchist (the ones defined something like "anarco comunist"), anarchist respect private property, and as such the integrity of a person (his body).
Actually you can derive a whole set of sound rules from this only principle (contrary to the democracy).
A difference with the usual system (the one we live in), that's not difficult to accept because it's quite natural
, so you just need force to defend yourself from aggressions, and not for enforce the rules, as in our society.
Free market = only a few economic rules, mostly just preventing fraud and colletting some taxes, the state still exists and acts on other things such as welfare, education, social disorder, etc.
If there is a super entity (i.e.: the state) that can unilaterally decide the rules, it's not a free market: for example, what about patents and copyright?