Rate is the price of a thing per piece, that can be multiplied to number of pieces to get price of the whole, it could be in terms of weight, length etc. whereas fee is a mandatory amount of money required to be paid for a certain service or an application for a certain job etc.

In my understanding they are fundamentally the same. However a rate, to me, sounds like something that is a bit more continuous, like a monthly bill, whereas a fee is a one time payment, such as a parking fee, contract fee (the fee paid to process a contract), etc.

Just to echo Bhaskar's point, rate is a ratio, usually of time dependent but not always. Fee is an amount of money for a transaction usually in addition to whatever the reason for the purchase, like a "service fee" is common surcharge for tickets. It is a bit sneaky in that an advertised unit price may be one value but there will be additional fees that raise the actual price of the item, again more common with services in the US.

Is there a certain context you are talking about? Like an interest rate on a loan?