Difference between "to foot the bill" and "to fill the bill"

To foot the bill is to cover the costs or expenses of something or someone. One of the meanings of the verb foot is to pay for/cover. Thus, expressions such as I'll foot it, foot the expenses, foot the costs, etc, are also valid.

foot verb (informal): pay (the bill) for something, especially when the bill is considered large or unreasonable.

^{— Oxford Dictionaries}

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• Not to worry. I'll foot the bill.
• That wedding must have cost an arm and a leg! I wonder how they managed to foot the bill.

To fill the bill means to be suited/fulfill all requirements. (Fit may be more common than fill for this expression.)

fit (or fill) the bill: be suitable for a particular purpose

^{— Oxford Dictionaries}

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• We need someone who is aggressive, motivated and disciplined. Do you fit the bill?
• He doesn't look like someone who fills the bill.

• fit (or fill) the bill: be suitable for a particular purpose (a partner is an ally or a companion, and you don't seem to fit the bill).
• foot (or pick up) the bill: (informal) pay the bill, especially when the bill is considered large or unreasonable.

^{form the New Oxford American Dictionary}

To foot the bill is an expression meaning to pay for something.

To fill the bill means that something satisfies a need or fits the desired specifications well.

Looks like lots of folks have mentioned how the two senses differ. I think it's worth pointing out that the word "bill" has several meanings in English, which ties into this. Consider these two meanings:

an itemized statement of money owed for goods shipped or services rendered; "he paid his bill and left";

vs

a list of particulars (as a playbill or bill of fare)

In the case of "foot the bill", we're talking about a statement of money owed (in many cases, a literal statement on paper).

For "fill the bill" the meaning is more like "matches this set of requirements", which relates to the second meaning of "bill".