Blizzard Time refer to the clock that you can have display at your map, showing how long the game has been going on. The clock can be enabled by going to options, gameplay, and then enabling the in-game clock.
At normal speed there is a 1 to 1 relation between Real Time and Blizzard time. But since most games are played at "Faster" speed it is usually not the case.
From Liquipedia page on game speed:
A rough guide for the differences between speed is below. This was
calculated using the 90 second MULE shut-down time.
Gamespeed # seconds to shut down % of standard time x normal speed
Slower 150 s 166% 0.599
Slow 112.5 s 125% 0.83
Normal 90 s 100% 1
Fast 74.5 s 82.75% 1.21
Faster 65 s 72.5% 1.38
However, due to the nature of deterministic synchronous game engines,
the actual real-time values cannot be calculated exactly. The game
attempts to keep constant 0.6x, 0.8x, 1.0x, 1.2x and 1.4x time
modifiers according to the crash logs, but due to network and
processing delays the actual observed time can easily shift 1-3% over
a given time period.
I would say the reason people want it to show Real Time is because of the possible shift and more importantly it would be easier to keep out of game timers for abilities and construction if you could relate the tool tip to real time even at Faster speed.