Hello
Lot's of things can be said but indeed, what was mis interpretated was the purpose of the French Mexican war : it was "just business".
Mexico was owing a very high debt to the European countries, so France decided to help in its recovery (mainly by looting Mexico's resources):
They invaded Mexico, gave power to a puppet emperor, took their money, and when the war started to be too expensive, they left.
There was no intent in staying long (unlike Algeria) and they wanted to spend as few money as possible to help for profitability.
Mexico resistance and US oppositions were only important as far as they made occupation more expensive.
It was a nice bargain for France and Europe in general.
The real mistake in France was that they were not concentrating on Prussia and did not realize their military organization was obsolete.
Edit : What is funny is that when you read french solidiers account on this war, you have the same feeling as when you read US accounts on Viet Nam war or ... Somalia
They said they won but that stupid politician made them leave the country and lose the war.