it might be a synonym but there is absolutely not the same subtext/tone, personally i would use 'extraordinary' as a much more natural feeling word, but like, supernatural carries to me strong implications there is an 'additional factor' in what gives them their power, it might not be magic specifically but it implies that a regular person wouldn't be able to do whatever it is they do without some kind of externally sourced facilitator, batman might be extraordinary in what he does but i don't think anyone would call his abilities supernatural.
edit: it's the difference between captain america and spiderman, cap might've had his STR, DEX & CON set at straight 20s but no amount of 'peak human potential' was ever going to give him spidey's wallcrawling or spidersense, extraordinary cap, supernatural spidey.