The most polite ways are probably: "May we have the bill/check, please?" "Could we have the bill/check, please?"
"Could we get the bill/check, please?
This has the meaning...
"Is it possible for you to give us the bill, as we're ready to leave and wish to pay"
Alternatively, when the server comes by and asks if you'd like anything else, a polite response would be: "
No thanks, just the bill, please" "Excuse me, Bill/check please" is casual and perhaps fine in casual situations, but it's still a little curt.
You can't go wrong with a full sentence question.
Note, in Canada, I've seen the word "bill" used most often.
Also note, in a Fancy restaurant, you will typically be asked if there is anything else.
A simple, "No thanks, I think that's all for tonight" will tip the server to bring the bill.
When paying, a credit card put in the envelope and left on the table will have the server silently take an bring back the completed bill.
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I think check is American, bill is British : both are fine, adding "Can I have" as Daniel says is certainly more polite.
A common way to ask for the bill is to just make extended eye contact with the waiter, perhaps making a 'writing a cheque' guesture.
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Two parts to your question.
Each part has been accurately but separately answered elsewhere but here's the summary: 1.
When asking for the bill/check in a restaurant, what's the polite way to phrase the question?
See Atomix's answer: "May we have the bill/check, please?" "
Could we have the bill/check, please?
" "Could we get the bill/check, please? 2.
Is there a politeness difference between bill and check ?
As TRiG's answer says, there is none.
But check is used chiefly in the USA and bill is used in the UK (and I think most other English speaking territories).
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"Can I have the bill, please."
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The word check should not be used in this context outside the USA.
In the USA, it's fine.
There's no politeness difference between bill and check .
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I don't think anyone cares which word you use, as long as you're polite.
I usually use the word "check".
Most servers interpret the "money" gesture (rub your index finger against your thumb) as "I'd like to pay now" and will not and bring the check.
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I used to say  check please , but my English teacher said that it's wrong, and the proper way is to say something like  bill please.  What's the truth?
How should I ask for a bill in a restaurant politely?
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<word-choice><nouns><politeness>
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The most polite ways to ask for a bill in a restaurant are probably "May we have the bill, please?" or "Could we have the bill, please?". You probably shouldn't use the word "check" in this context outside of the USA. "Check" is American, "bill" is British : both are fine, and adding "Can I have" is certainly more polite.  
The most polite ways to ask for a bill in a restaurant are probably "May we have the bill, please?" or "Could we have the bill, please?". You probably shouldn't use the word "check" in this context outside of the USA. The most polite ways are probably:  "May we have the bill/check, please?"  "Could we have the bill/check, please?"  But check  is used chiefly in the USA and  bill  is used in the UK and  most other English speaking territories.  Being polite is the main thing.
A common way to ask for the bill is to just make extended eye contact with the waiter, perhaps making a 'writing a cheque' guesture.  
The most polite ways may be "May we have the bill/check, please?"; or "Could we have the bill/check, please?" The most polite way to say this is probably "Can I have the bill, please?" or "Could we have the check, please?". Alternatively, you might find that making eye-contact with the waiter and making a writing gesture will be enough.
The most polite ways are probably:  "May we have the bill/check, please?" OR  "Could we have the bill/check, please?"   Check is American, bill is British : both are fine, adding "Can I have" is certainly more polite.   "Can I have the bill, please." The word  check  should not be used in this context outside the USA. I don't think anyone cares which word you use, as long as you're polite.
The most polite ways are probably:  "May we have the bill/check, please?"  "Could we have the bill/check, please?"   But check  is used chiefly in the USA and  bill  is used in the UK and  most other English speaking territories. A common way to ask for the bill is to just make extended eye contact with the waiter, perhaps making a 'writing a cheque' guesture.   Being polite is the main thing.
The most polite way to do this is probably to ask "Can I have the bill, please?" or "Could we have the check, please?". Making eye-contact with the waiter and making a writing gesture with your hand may be enough.