To start with - and not to pick on you or anything - The Tudors isn't a good historical source, so you don't want to assume that an aspect considered "French fashion" on the show is in fact historical French fashion. The costumes are all over the place, quality-wise and in terms of what time period they actually depict.

French fashions are frequently connected with Anne Boleyn, who is said to have brought them back with her from the French court, but we don't actually know very much about what they looked like. (Re: Anne Boleyn, this seems to me to be more of an attempt to tie changes in historical fashions to important women than a well-supported argument.)

One thing that we do know is that there was a French hood and an English hood. The English hood, also called the "gable hood", is essentially a box on top of the head: see Mary, Lady Guildford - (https://commons.wikimedia.org/wiki/File:Hans_Holbein_the_Younger_-_Mary,Lady_Guildford(Saint_Louis_Art_Museum).JPG. (Earlier, the sides were very long, hanging on the chest; over the years they were folded up and then began to be cut at the level of the jaw.) The French hood is essentially a multi-layered veil, with each stripe visible (as in this portrait of Catherine Parr) representing a different piece placed on top of and slightly back from the edge of the previous layer.

It's the French hood that's usually specifically tied to Anne Boleyn - mainly because it makes a good story. Catherine of Aragon was depicted in an English hood in many portraits, while the famous posthumous one of Anne shows a French hood, and Jane Seymour was also painted in an English hood. The English hood looks dowdy and repressive, while the French hood is smooth and pared-down. Thus we get the relationships shown symbolically: Catherine was conservative and unattractive, Anne was pretty and forward-thinking, and Jane was repressed. This is so convenient as to make the "French hood suddenly introduced by Anne" narrative highly suspicious. Other suspicious things include the fact that Mary Tudor wore one in England long before Anne Boleyn was a player; Catherine of Aragon also may have worn the earlier form of the French hood as well when Anne was still a baby. (I say may because that portrait might not be Catherine - but it's commonly attached to her.) There are also two portraits of Anne in which she wears a gable hood. So.

Apart from that, we do have some inventory entries that refer to gowns with "French sleeves" or "Italian sleeves", but as they aren't illustrated we don't really know what that refers to. Portraiture doesn't show striking differences between French and English fashion during this period.