Amsterdam to Munich

Short comment: Don't try to see too many cities into a week, not more than 3 in my opinion.

Cologne is not that attractive unless you are seriously into medieval churches. Have a look at Hamburg - you might well find there's enough there and in Amsterdam to spend a week. Alternatively Berlin would be good.

If I am to stay between Berlin and Hamburg what towns or places would be good to visit for both history and art.

"Neuschwanstein Castle would be wonderful"

Besides it's location hundreds of miles away - which you would travel for a 30-minute tour?? - it's not a genuine castle - just a newer palace with a fake castle exterior.

Near Hamburg: Look into Bremen. Also check out Lüneburg for a look at smaller old-world Germany.

A'dam, Hamburg, Bremen and Lüneburg is probably way more than you can do in a week. Fly out of Hamburg if possible. Or perhaps make the trip back to A'dam to fly out if the price is right.

What periods of history and art are most attractive to him? I remember being quite impressed by Cologne's Roman museum on my first visit, but current remodeling means that exhibits have been relocated. If 20th century art is of interest, then the Museum Ludwig next to Cologne's cathedral and main train station is an easy stop.

She is interested in anything related to older history and artists. As she puts it she lived Ancient Egypt, then resided in Versailles, She remembers Van Gogh and Abraham Lincoln as well as JFK. She has a block and will not relive past the '80s. Yes, I am very serious. So anything that is History fascinates her. We are going to Hamburg specifically for the Beatles but know Germany has so much more history beyond them.

Castles: The Marksburg on the Rhine, which is a real one:

https://www.marksburg.de/en/visitor-information/

Amsterdam to Rhine valley (St. Goar where you can catch the ferry across to the train on the east side of the River); Rhine Valley to Hamburg, Hamburg to Berlin. Fly Home.

Have a great time.

Besides it's location hundreds of miles away - which you would travel
for a 30-minute tour?? - it's not a genuine castle - just a newer
palace with a fake castle exterior.

How do you define "a genuine castle"?

"Castle" - For castles in Germany, the German definition applies, and by definition, they were built in medieval times for purposes of defense. See "Burg" below.

https://www.duden.de/rechtschreibung/Burg

In Germany, a "Burg" is routinely confused with "Schloss" (palace) by outsiders whose native tongues and cultures are unfamiliar with palaces and castles. Both words exist in English of course - but most English-speakers, new-world English-speakers anyway, have no exposure to these buildings and no firm understanding of the distinctions, so in actual speech, their words and definitions tend to be loose and somewhat interchangeable. Thus "castle" gets used incorrectly by tourists in Germany who end up at Neuschwanstein and similar buildings which are not genuine castles.

Not so in Germany, which is home to thousands of castles and palaces. German kids learn the difference early on:

https://www.t-online.de/burg-oder-schloss-was-ist-der-unterschied-/id_67383670/index

Here's a page in English that explains the differences: https://www.familyhandyman.com/smart-homeowner/difference-between-castle-and-palace/

I do find it interesting that Germans do not define Neuschwanstein as a castle. But from the many responses, I am considering staying in the Northern region, just looking at places to visit. I think she would like Cologne for the Medieval aspect but may just check our luggage, stay a few hours and move on to another town to stay the night. I had also considered flying into Munich, then doing Munich, Heidelberg, a quick stop in Cologne, then Hamburg and Berlin but that may be too ambitious as well for the time we have. If I can find enough interesting spots for art and history in the northern region I would just stay there and avoid travel.

@Russ: Thank you! I speak German so I'm well aware of the differences, but this is probably the first time I've seen an american aware of it. Other languages also have a lot of different words too. But my impression have so far been that in english (at least in US/Canadian english) everything is lumped together as castles.

"...everything is lumped together as castles."

Yes, the English word covers a lot of linguistic territory. I think part of it is just our playfulness with words in America, where "a man's home is his castle" is a common piece of wisdom, where Walt Disney builds "castles", where Steve Miller's Billy Joe, song character in "Take the Money and Run," "shot a man while robbin' his castle." In Europe, it's no wonder that we seem a little dopey about the real buildings.

But it cuts both ways. Sometimes we differentiate where Europeans might not. Lots of European visitors in California don't seem to grasp how we differentiate between the uses of "freeway and highway", for example.

A Swiss friend of mine was once fairly upset that ann American diner, where she had ordered a hot beef sandwich, had the nerve to serve her a messy, steaming pile of gravy on top of sliced beef on top of plain white bread. "Now just how is that a sandwich?? We wouldn't call it a sandwich in Switzerland!" Still gives me a small chuckle.

Since you are up north, consider going to Lübeck or Quedlinburg.
For a magnificent palace, go to Potsdam and visit Sans Soucci, rather than Neuschwanstein 100's of km away from where you are. Dresden is also not too far away and would have a lot of sites to interest your daughter.

In Europe, it's no wonder that we seem a little dopey about the real
buildings.

You don't. It's understandable when you don't encounter these buildings regularly, there is really no need to tell them apart. And it doesn't help that some buildings don't really fit in one category or another since they have been rebuilt during the centuries and used for different purposes. But if castle is the equivalent of Burg, is there an umbrella term for all these kinds of buildings?

"But if castle is the equivalent of Burg,"

As it is found in everyday speech, castle is not "Burg." It serves most often as what you all an "umbrella term." But in actuality, everyday English doesn't really have two separate real-world referents for "castle" since speakers make no distinctions when they use or hear the word. Most people don't talk much about such buildings and have never thought much about castle vs. palace or may not have cared to unearth the reason for having 2 different words.

More "official" English sources like dictionaries usually attempt to keep the distinctions alive - to preserve their traditional meanings. These usually don't consider "castle" an umbrella term, but a "Burg" in the German sense of the word.

Sources with more a more "descriptive" focus - the ones that only describe the language as it appears in actual use - may embrace it as an umbrella term, including a number of different possible uses.

AFAIK there is no "official" umbrella term that shelters the traditional definitions for both "castle" and "palace."

Anyway, I think any English speaker who travels to Germany - the place with all the palaces and castles - ought to be cognizant that Germans view the two types of building as essentially different and use different words for them - in the traditional manner. Part of travel to Germany is understanding differences in the way Germans view the world. And that sometimes involves examining and crossing over the borders of one's own spoken native language.