I plan on going to Tokyo, Japan or Taipei, Taiwan in late September or October 2015. Any suggestions as to which of these places has more interesting things to do during this time of year?
Both countries are great and I only spent some time in August in Taiwan but there was a lot to do for a week I would say. I write a bit of a travel blog, and I wrote a post about Taipei Sun Moon Lake Hualien City in Taiwan as good places to visit, so check it out :) http://bit.ly/1CXciQ5
I lived in Japan for one year and Autumn is a great time to visit, but more so October than November as this is when all of the leaves turn fantastic colours and its great to see! If you plan on going to Japan then the Kansai area is great this time of year, especially Kyoto Nara and Osaka. Here is another post on the great places in Kansai to visit:
Kansai region and things to do: http://bit.ly/1EFoek8
Kyoto and good places to visit: http://bit.ly/1cU5dnl
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The answer would really depend on what exactly you are looking to do as both are nice at that time of year. There is a lot to do in both cities, but you could probably occupy the entire two weeks in Tokyo and not see it all. Taipei would be great for two weeks but you'd also be able to see a good portion of the city and travel outside to some of the gorgeous coastal areas of Taiwan with enough time. The two countries happen to be my favourite :)
Taiwan is cheaper in general then Japan so that may be a factor that is important to you. It's MUCH cheaper in Taiwan for transportation then it is in Japan - you'll be able to see outside of Taipei for a comparable price to getting from one end of Tokyo to the other. Accommodation is cheaper as well, depending on where you plan to stay, and you'll find a larger variety of night markets and cheaper meals in Taiwan.
Night life is different in each - you'll find the nightlife lasts longer in Japan overall and there are more things to do, but having said that, much of the activity we saw was from local Japanese people. Taipei has a large expat community that's pretty centralized - you'll be able to find several bars, clubs and lounges open late that have both Taiwanese and many expats/foreigners.
There are beautiful natural sights, hikes and activities, onsens/hot springs in both countries, but like I mentioned, it'll cost more to get around to see them in Japan.
Are there specific things you are looking to see and do? Attractions or activities you are more inclined to participate in while travelling? Taipei is a big city at a much slower pace, it's calmer then Tokyo and significantly less crowded.
We've written a bit about Taiwan and Japan (http://onemoderncouple.com/category/asia/japan and http://onemoderncouple.com/category/asia/taiwan ) and there are some videos that sum up our travels to each, but a lot of our information is not yet posted, so please feel free to follow up with more information and I can try and help narrow it down further!
I don't know about Taiwan but late September and October are beautiful season in Japan.