Answer:
There are many, and as Spanish and Mexicans get to know each other more, the quote above rings truer. Although the shared language and 300 years of shared history keeps us very close and makes it so we understand each other readily, we tend to argue on the details when we’re more similar, but become more open minded about each other when we’re different.
Mexico is the “ultimate cultural appropriator”, so it has accepted cultural things from more places than Spain. Here are some differences, classified.
As a rule of thumb, anything Mexican that comes from these cultures is a difference:
Americanized stuff - Modern cowboy Tex/Mex mixture (although the old vaqueros were indeed from Spain), pop music, etc (although since the Spanish “revival” after the Franco years which brought its own American influence, we have merged a fair amount of that directly though similar “Rock en Español” productions).
French Stuff. Mariachi, Republicanism (including parts of our Constitution, and also ours succeeded, theirs failed, and there’s a lot to talk about right there), Liberalism, Revolutionary influences. We got all that from the French, not the Spanish. A lot of our 1860–1920 architecture is also evidently French.
Mexico is in Latin America, while Spain is in Europe, and is part of the European Union. Spain is more industrialized than Mexico, and the rule of law is stronger there. Mexico is still in some ways a developing country, and beset with problems of violent crime, and powerful drug cartels.