June or September 2020?

I'm planning on visiting Japan next year for around 2 weeks and have a couple of options for time of year either:

Early to mid June or

Mid to late September or even into mid October

Consider both weather and the Olympics, what would be the best time of year to visit?

Early to mid June.

Depends what part of Japan you're going to....?

Assuming you do not go to Hokkaido but somewhere else, for me there is no question at all: September.

In June you get rainy season and once that is over it will be brutally hot and humid.
In September you can get typhoons, but they are one day events and temperatures have dropped. Specially from mid September onwards. (Dropped simply means that they have become bearable. It�fs still warm).

Enjoy planning your trip to Japan!

You mentioned October. If you can do late September into October, even better.

We plan to go to Tokyo and Kyoto and maybe Osaka for a day or so.

The average temp for June appears to be only ~24 for Tokyo but I do keep hearing it is very hot so a little confused!!

My experience of June in Tokyo is that it is blistering hot and quite humid. But then if you crack a few rainy days it can bring the average down. I just looked at the daily temps in Shinjuku in June 2018, most are 28 - 33 celsius highs but there are a few from 18 to 24 highs. Being among the glass buildings, it really gets stifling.

Personally, I wouldnt go to Tokyo or Kyoto in June, both are like Satan's armpits. Hakone, Karuizawa, Kamikochi, Nara, Bessho Onsen, Nagano are central areas which are quite nice that time of year. I'm sure that there are plenty more.

September/October.

Last month (June 2019) in Tokyo wasn't too bad at all, although there were a few very very hot days. In my experience, the first few weeks of September are much hotter.