Туфли - either women`s high heels or men`s shues that go with a business suit. Ботинки - shoes for autumn or winter The name for all shoes is "обувь". For exapmle, летняя обувь, зимняя обувь, мужская/женская/детская обувь, etc. Check "туфли" and "ботинки" in Google Images, you`ll see the difference

Besides I might be wrong about what is more common:( It is just my usage. Besides-2 when you ask "didn't you see my ....?" you may decide to be specific.

Also in shops they sometimes formally distinguish туфли, полуботики (half-boots) and ботинки. We are unlikely to say say 'полубоинки' unless in the context of shopping though:) Now, for me 'ботинки' is the most 'simple' word in a sense. Closer to 'archetypical footwear'. So my shoes is "мои ботинки" is this is men's shoes... and "туфли" if this is women's dress shoes (women's dress shoes look TOO much unlike the mean between 'shoe' and 'boot':)) But don't be surprised if somebody says "мои туфли" (just because he likes the word) or if I say штиблеты.

туфли - a range of things. First of all they are rather low. Dress shoes can be called this way. We mostly associate dress shoes with this name now (i mean - the first image to come to my mind is rather smart italian made something:)), but in past ot was appied to many things, including soft things worn in home, tennis shoes and to sneakers. Well, actually the term is still used this way, in some shops they sell sneakers as туфли.

"Кроссовки" (< кросс (cross country running)) = athletic shoes, running shoes tennis shoes etc. We would call some kinds of hiking/trekking shoes this way too.... But when they are high/heavy/otherwise impressive enough they may evolve to ботинки:) Кеды - sneakers. Key features are thin sole and thin everything and... well,they normally have classic design.