A lot of people will say England because of the language. Well, I spent a weekend in London and never understood anyone. So many people in England speak dialects that are very difficult for American's to understand. On the other hand, if you stay in well traveled tourist areas in Germany, there will be people there who speak English (about 60% of Germans do) and they are taught the "Kings English", which is a lot easier to understand than English dialects. I also feel that the culture of Germany is more similar to ours. So, I'd go to Germany.
I like David's suggestion of Munich. If you like museums, there are plenty to see in Munich. There are art museums - the Alte- and Neue-Pinakotheke and the Pinakothek der Moderne. Right in the same area, a U-Bahn stop from the Hbf, is also an Egyptian museum, a Paleontology museum, and a museum of Greek and Roman sculpture. Out on an island in the Isar River, east of downtown, is the Deutches Museum, their version of our Smithsonian.
Unless I wanted to go somewhere really, and I mean REALLY, out of the way, I would never rent a car in Germany. Germany has the most extensive rail network in Europe, 40% larger in rail km than that of much larger France, and only 1% less dense than the much smaller Swiss network. With almost 1000 stations in Bavaria alone, you are never far from a station, and the rail system is enhanced by a vast network of public buses.
As for the Romantic Road, as has been mentioned here multiple times, there is nothing exceptional about the road itself, a two-lane country road clogged with tour buses, lorries, and farm equipment. It's the towns themselves that make it interesting. In 2007, I did the entire Romantic Road, except for Füssen to Weiskirche, entirely by public transportation (it can be done). I would say the best town on the Road is Rothenburg, and it is readily accessible by train. However, everyone wants to go there; some think it is too touristy. A less touristy version of Rothenburg, complete with a mostly intact wall with a Wehrgang you can walk on, plenty of Fachwerk buildings, and a church steeple you can climb, is Nördlingen. Nördlingen is also readily accessible from Munich by train (in about 2 hrs). From Nördlingen, Dinkelsbühl is less than an hour away by direct bus, but I found Dinkelsbühl a disappointment after Nördlingen. If you stay a few nights in Nördlingen, an interesting side trip by train is about 15 minutes south in Harburg, with it's intact castle.
From Munich, you can go by train to Salzburg or Berctesgaden with a stop on the way at Ludwig's island palace of Herrenchiemsee. Or take the train via Mühldorf to Burghausen to see the castle there.
For a mountain experience, take the Meridian train to Brannenburg on the Inn River. From there take the Zahnradbahn (cog railroad) to the top of the Wendelstein. Then come down the Seilbahn (cable car) on the other side to Osterhofen and take the Bayrische Oberlandbahn (BOB) back to Munich. Of course, if you tried to do this by car, you'd have to backtrack to Brannenburg to pick up your car. If you leave Munich at 8:44, you'll get to Brannenburg at 9:40 where you'll be met by the Wendelstein Ringlinie bus at 9:45, which tales you to the Zahnradbahn Talstation at 9:51. Otherwise it's a half hour, 1½ mile walk to the Talstation.