Japan is finally at the top of my list - but should I go to Korea instead?

Both countries are beautiful and are worth a dedicated visit. It really comes down to your particular interests, your budget and your available time. For a first time visitor a tour such as you propose is probably the best way to get around. Going independently is doable but with no Korean language skills it would be a rather daunting experience. Seoul is a modern metropolis with all of the positives and negatives that go with that - the traffic being the biggest negative in my opinion. Another reason to sign up for a tour I suppose.
The countryside outside the cities is beautiful and the country itself has a truly fascinating history. For example, most people aren't aware that WWII for the Koreans started in 1910 with a Japanese occupation - an altogether unhappy experience for the local people (who still hold a grudge after all these years).
If you can arrange it I'd suggest finding a tour (or an addendum to one) that would get you down to Jeju Island for a couple of days. It's a little bit of Hawaii in Korea, with beautiful beaches and seascapes if you're interested in that sort of thing.
Japan is terrific too. The ancient city of Kyoto is worth several days by itself, and getting around the country independently is surprisingly easy. The rail transport system is world class and English subtitles on the signage makes it a breeze to navigate.

Can't help you with Korea, haven't been there yet. But for Japan, I'll just add this: I see no need for any kind of a tour there. It is perhaps the easiest foreign country I've been to for someone who doesn't speak the language and can't make any sense of the written language. The country has been getting lots of foreign tourists for a long time and it shows - they know how to make things easy for clueless outsiders. Even if you hit a language barrier (not everyone speaks English) just about everyone is incredibly polite and helpful.

Thanks for the input so far. @David - this will be a tour - for a variety of reasons. I usually travel on my own...but sometimes a tour makes sense and this is one of those trips.

As to which country - I had always assumed it would be Japan but recently I came across Hana Tours and the tours look pretty good...they're hectic as they move around almost every night, which makes me a little nervous. But they are a great deal and they're jam-packed with activities. Groups are small - minimum required for departure is 2 and TripAdvisor reviews indicate that most groups are under 10.

I was just curious if anyone on here had traveled Korea. The Gate 1 tour prices out at $3500 for 10 nights. Hana Tours is $1900 for 9 nights. And I save about $500 per person on air for Korea vs. Japan. So savings per person is $2100.

Can’t help with either destination, but just want to point out that tours that are jam packe with activities and spend only 1 night at a hotel are exhausting and rushed.

I tend to look for tours that are two to three nights in one place because I’ve done the one night tours in the past and find it gets tiring to repack and check out every single day.

Not suggesting you don’t do the tour you want, just wanted to provide another point to consider.

Do them both and pay the trans-Pacific airfare just once?

Noah Smith has a very good info on touring Japan on his blog:
http://noahpinionblog.blogspot.com/2018/06/noah-smiths-japan-travel-guide.html

If you have the time to combine both countries on a single trans-Pacific fare, know that both Seoul and Tokyo are served by domestic "city airports" that are located much closer to the metro areas than either of the major international airports at Incheon and Narita. The one in Seoul is Gimpo (GMP) and the one in Tokyo is Haneda (HND), and there are several daily non-stop flights connecting the two. Both are easy and short taxi rides into town... much easier than using the international airports anyway.
Doing your transpac as an open jaw (flying into Narita and out of Incheon, or the reverse) and then using the two city airports to connect them would be a cost effective and convenient way to visit both countries on one ticket if you wanted to combine the two.

Will just point out that the move everyday tours are really exhausting and usually you need a vacation from your vacation. Plus you really don't get to see that much being so rushed. Try to find a tour that stays two nights in some cities. As for which to visit, a family member has been to both Korea and Japan. While Korea was interesting, they loved Japan for the cleanliness, politeness and wealth of things to see. Yes, it's expensive but there's just something about Japan away from the big cities.

South Korea's mountains got my interest during the Olympics. I want to ski there!! That said, I lived in Japan and could not recommend visiting enough. I think whichever you choose, you will have a wonderful time!

If you're American, there is a travel ban to North Korea instituted after the North Korean government 2 1/2 years ago incarcerated an American student, Otto Warmbier, and sentenced him to 15 years of hard labor under accusations that he stole a political poster while on a visit to North Korea from his study abroad program in Hong Kong. Under this imprisonment he went into a coma never to come out, was finally sent home, and died in the US after 17 months in captivity.

So "Korea" tourism for you means South Korea.

Beside the most well-known destination Jeju island in South Korea, you also see amazing scenery in SeryangJe too.
I like Japanese culture too. As I know, most tourists are really happy after the trip. In japan, what I can list to you is Kawachi Fuji garden in Fukuoka, Koya lake in Niigata (so beautiful in autumn), Otaru festival in Hokkaido. It's so beautiful in spring, when cherry blossoms blooms with their gentle, delicate color.

We loved Japan, it was clean, tidy and the people are friendly and so helpful. Also, there is a lot to see there. Kyoto is amazing, it wasn't bombed in WWII.

We haven't been to Korea, but my Son visited there for a week and said it was as interesting as Japan.
Also, my Daughter served there with the US Army for a year and loved the country and its people.