Both cities have affordable options. Just follow the university students and they have a knack of finding cheap eateries. While Kyoto doesn't have the street bars and izakayas underneath the train tracks like Tokyo, try the back streets for low cost Japanese food, like noodles or okonomiyaki for 500 to 800 yen, http://en.japantravel.com/view/tsutsumi-okonomiyaki-house or head to the more working class areas one stop south of Kyoto Station on the subway, or head towards the Hankyu line or the tram stops near Omiya Station. Several floors on top of Kyoto Station is a ramen alley, which is a sanitized version of an eat street.
For accommodation, try http://en.japantravel.com/view/khaosan-kyoto-guesthouse. They have a branch in Tokyo as well, for around 3000 yen. There are places to go for free as well. http://en.japantravel.com/view/kyoto-for-free
I would have to say Tokyo. With its sure size, there are many more options for cheaper food, hotels, and shopping. Transporation is cheaper in Tokyo if using trains, but Kyoto is very doable by rental bike--unlike Tokyo. Both cities will have some free attractions as well. You must factor in the cost to get to Kyoto and back if you are flying to Japan from a western country. You will most likely land in Narita or Haneda ariports near Tokyo.
We sometimes stay in Kyoto for one or two nights and the accomodation isn't too cheap even if it's a medium-quality hotel. It is much similar to Tokyo. Restaurants are usually not cheap, too. Once we stayed in Nara, it was twice cheaper than Kyoto. Visiting shrines and temples in Kyoto, expecially in November and March will take some more money out of your pocket.
I think what helps both cities become more affordable is the great selection of hostels and guesthouses that have opened recently in both cities, too many to even list here. Before, I feel Kyoto was much pricier on the accommodation side but thanks to unique capsule hotels like 9Hours and Capsule Ryokan and guesthouses like Gogo and Rakuza, the city becomes much more affordable. Food-wise, however, Kyoto might have a slight edge, but only because I feel there more much more cheap eats clustered around all of Tokyo's numerous train stations, while Kyoto doesn't quite have the same scene.
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