As the title said, Do these 2 sentences have the same meaning?
Both mean the same and are grammatically correct but 'live without rules' sounds better.
Yes, the sentences mean the same. However, the first one can be (but does not necessarily have to be) used to emphasize the fact that there are no rules, while the second one is a statement without real emphasis.
The two formulations mean the same. I would say "without" is the normal thing as it is one word and shorter.
For all practical purposes, an English language learner should consider "living with no rules" and "living without rules" to be synonymous.
I could just stop there, but I should mention that there is possibly a very subtle distinction. "Living without something" can have an additional connotation that the something is too expensive to be worthwhile. Illustration:
"You're telling me that the car's built-in GPS will cost $500 to fix? In that case, I'll live without it."
Meaning: the GPS will probably remain in the car, not working, because I don't want to spend $500 to fix it. For that usage, "I'll live with no GPS" would not be quite as good an expression, partly because the GPS unit is physically still present, and partly because "living without" can have the additional sense of "surviving or making do with a worse option".
So, you could say: "If the only way to make rules to control noise in the neighborhood is to form a homeowners' association, then we'll just live without the rules". It wouldn't be quite right to say "we'll just have to live with no rules", since it's still not a lawless society.
I could just stop there, but I could go yet further and mention that "living without the rules" could have a completely different interpretation altogether. Without can be used as the opposite of within, though such usage is archaic or literary. Therefore, it is possible to interpret it as the opposite of "living within the rules", but that is a very twisted interpretation — probably one that no sane person would think of.