As Matt mentioned, the handwriting changed rather dramatically. Until the mid-20s, children were still taught a form of German cursive script known as Kurrentschrift or now often called Alte (deutsche) Schrift. The letters are all slanted and are specific to this form of writing - a written alphabet particular to German since before the 15th century (which means, incidentally, that if you can read that script from the World Wars, chances are good you'll also be able to read letters from the 1700s fluently).
This script was "modernized" in the early 20th century into a form called "Sütterlin." The letters remained similar, but now were all upright rather than slanted. This form wasn't particularly popular and was dropped again pretty quickly. You'll encounter this mostly in period primers from school and the occasional government form filled out by someone who went to primary school during that window.
In 1941, it was decided that Germans should standardize their printed and written word to conform to European-wide practices - the latin script. You'll notice in books, government documents, magazines, newspapers, etc etc that the Gothic script gets dropped in the course of this year and gets changed to latin type. The handwriting taught in schools also changed to latin cursive, the type that is still taught today in Germany (and for the most part looks like the cursive we all learned in school too). As in the case of your letters, though, of course the older generation mostly continued writing in the old script - same for most documents and forms you'll encounter. Interestingly, a common practice for POW camps was to force the German inmates to write in latin cursive or block letters to make the censors' jobs easier. As a result, these letters home often look like they're written by children. For many of them, it's the first time they've had to use a handwriting other than the old script.