What are the biggest differences between the American system of movie ratings and European systems?

Wikipedia has a comprehensive chart comparing how different rating systems around the world are structured.

One important difference between MPAA ratings and some other countries' rating systems is that the MPAA is a non-governmental organization and its ratings are voluntary, whilst in many countries ratings are mandatory and/or given by a governmental agency; for example, France's Ministry of Culture is in charge of ratings in France. In the United States it is not a crime to permit a child of any age to view any non-pornographic film, whereas in some countries admitting children to films of certain ratings is illegal.

It's hard to compare the criteria other countries use against those in the U.S. because the MPAA is famously opaque about what criteria they use. For a deep dive on this topic, I recommend the 2006 documentary This Film Is Not Yet Rated. Moreover, what criteria do exist have changed over the years: early G ratings like Head and Tora! Tora! Tora! contain graphic war violence, whilst the formerly X-rated Midnight Cowboy was later reduced to an R rating without editing.

Broadly speaking, sexuality and coarse language are more likely to inflate a rating in the United States than elsewhere, though this is of course a gross generalization. Conversely, certain types of content are treated more seriously outside of the United States, such as "incitement to violence" and "incitement to hatred", and it is difficult for any film to receive an NC-17 rating from the MPAA on violence alone. These countervailing forces have resulted in an overbroad R rating that includes films with merely moderate coarse language alongside films with extremely violent scenes and adult themes. Some European rating boards tend to be more consistent and transparent.

Slightly off-topic, my favourite MPAA rating is Twister's PG-13 "for intense depiction of very bad weather". Somehow, The Wizard of Oz, whose tornado terrified generations of children, received only a G rating.