The two appear to get different results. To get up at 7:00 AM, sleepyti.me says to go to sleep at 10:00 PM, whereas jollynap says to go to bed at 10:45 PM. Obviously the latter is taking 15 minutes into account for falling asleep, but it's still an hour difference.

SleepyTime mentions, "The average adult human takes fourteen minutes to fall asleep..." so I compensate with 15 minutes in the calculation. There shouldn't be any major variance. Perhaps an odd calculation with Daylight Savings?

The sleep now option doesn't seem to take my time zone into account.

The time is based on your browser's Date function which should match your system clock. I think you might be experiencing a strange calculation with Daylight Savings.