In google these two sentences were both translated into: Can I pay by card?
Whether there exists any subtle differences? Or are they completely equivalent?
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They are not equivalent. The difference between the two is that of ability vs. permission. In English, both of these concepts can be expressed using "can", which is the reason why Google Translate offers the same result here. So these two sentences actually mean:
(1) Am I able to pay by card?
(2) Am I allowed to pay by card? / May I pay by card?
I suppose that in this context, the difference in practice is not very big, but in other cases it definitely can be. For example:
(1) 食べることができますか。 Am I / Are you / Is he etc. (physically) able to eat?
(2) 食べてもいいですか。May I eat? / May we eat?
カードで払うことができますか。 am i able to pay with a credit card (at this store)?
カードで払ってもいいですか。 may i pay with a credit card (at this store)?
I agree with Kaskade's comment. Ability vs Permission. In the case of the credit card example, you are able to use 払うことができますか because the store may not be physically able to process your credit card (lack of a credit card machine). It's not a matter of permission. It's a matter of ability.