"This web site is intended as a time-of-day service only. It should not be used to measure frequency or time interval, nor should it be used to establish traceability to NIST or the USNO."
"REAL time" is a bit of a misnomer. It all depends on how well you can realize the definition of the second, and transmit that information. Over the internet, that's limited to about a millisecond with NTP (Network Time Protocol)
Anyway, the two sources in the US, as the site says, agree to better than a microsecond. But the USNO follows the BIPM (the International standard) a little more closely. But then, we contribute upwards of half of the data that they collect.
edit to add: and I always chuckle at the mention of "the" atomic clock. The time that gets disseminated from USNO is the averaging of dozens of atomic clocks.