The two words are indeed different so let's first look at the word 'fee'. Fee usually describes the amount of money charged for some professional service like lawyer's fees, or professional advice like counselling fees. Here are a few examples:

International school fees have risen by 5% this year.
I have to pay the solicitor's fees for helping me make a will.
What are the fees for a weekly house cleaning service?

Another example is when you register for a course; there may be a registration fee which pays for your application processing. Likewise if you were to use an agent to help you process a visa application, there would be an agent's fee as you have to pay for their help. Sometimes (well, in the UK at least), a plumber or an electrician will charge you a 'call-out fee'. This is an amount of money that you need to pay just for them to travel to your home and assess the problem. Also if you go to a bank to change money from one currency to another, there may well be a 'fee' for this service.
So what about 'fare'? Well, generally speaking, 'fare' is only used for transport. A fare is payment for a trip on a bus, ship, taxi or train. The passenger pays a fare on public transport. So it is correct to say 'bus fare' but not 'bus fee'. Some examples for you are:
What's the bus fare from here to the city centre?
If you travel between 10am and 3pm, you can get half price train fares in April.
I don't have enough money for the taxi fare home.

fee: 服務費, 酬金(如付予私人教師﹑ 醫生等的)
pay the lawyer's fees
付律師費
a bill for school fees
學費帳單

fare: (公共汽車﹑ 輪船﹑ 計程車等的)票價
What is the bus fare to London?
到倫敦的公共汽車費是多少?
travel at half/full/reduced fare 半價/全價/減價/票旅行