For books, the two I would recommend that are generally available are:

Claude Blair - European Armour Circa 1066-1700 (available from well-stocked libraries or online sellers)

Edge and Paddock - Arms and Armour of the Medieval Knight

They are both general introductions, and Edge and Paddock honestly take so much from Blair that they are fairly similar in content, though Blair is more focused (Edge and Paddock have better pictures, though). Tobias Capwell at the Wallace Collection has written a book on English Armour 1400-1450 that should provide much better information on English helmets specifically. In addition, a thorough online collection like that at the Philadelphia Museum of Art and the Metropolitan Museum of Art can show you different styles of helmet if you search by country of origins, though this is more of a supplement to the books above.

That said, some generalizations, focused on the 15th century, my main area of study:

Types of Helmets:

Bascinets - 14th and early 15th century - Used throughout Europe. The main regional difference is in the way the visor is attached - the Germans use a single pivot in the center of the helmet (a 'Klappvisor') after the Italians and others have moved to a side-pivoted visor in the later 14th century.

Great Basicinets - (IE, those which replace the maille aventail with metal plates) - 15th century, most popular in c1420-1450 or so - Primarily used in Germany and England, France and the Low Countries; Italians seem to prefer armets to great basinets in the second quarter of the 15th century. German examples are close fitting to the skull, while examples from England/France/Flanders etc. are bulkier.

Armets - 15th century - in the second and third quarter of the 15th century they seem to be most popular in Italy, so earlier armets are probably Italian. Toward the end of the century they seem to be more widely used. Flemish and Spanish armets in particular often have bellows-like fluting on the visor.

Barbutes - 15th century - These helmets (with their t-shaped face opening) are almost always Italian.

Kettle Hats - 13th-15th century* - These are widely used, though I see limited evidence for the English using them very much, at least after 1400. Some types of German kettle hats had a brim turned around evenly all around the head with sallet-like vision slits; other nations didn't seem to use these much. Many 15th century kettle hats have brims turned down at the sides but up in the front and back (kind of like a later morion); the Spanish in particular seem to have favored this with a bevor that covered the entire face, even including vision slits within the bevor (many examples are illustrated in 'The Medieval Armour from Rhodes' by Karcheski and Richardson).

Sallets - 15th century - Every nation in Western Europe used sallets. They are nearly ubiquitous from 1450-1500.

Sallets in Germany - Generally, the Germans were later to adopt sallets and eventually developed sallets with very deep skulls and very long tails (occasionally the tail is articulated). Some 'black sallets' are left rough from the hammer and often painted. Around the turn of the 15th-16th centuries Sallets with very deep skulls, short articulated tails and visors that cover nearly the entire face appear, often with both vision slits and breath-slits near the mouth (the overall effect is that they look like Iron Man).

Sallets in England/France/Low Countries - Generally these are less deep than German sallets, with somewhat shorter tails. Sometimes made in Italy for export.

Sallets in Italy - Toward the end of the 15th century deep sallets with bellows-shaped visiors appear; these are almost always Italian.

Close Helm-Sallets - 1480-1500 - Sallets with a bevor that is attached to the helmet and can pivot up. These are generally from England or the low countries. EDIT - However there are some German examples, following the broader German sallet styles.

*Even later if you include Morions.