Tarkin is concerned that the Death Star won't work, given that it's behind schedule and beset with problems. The last thing he wants is for Vader and the Emperor to schlepp across the galaxy only to have it go fzzt and not do anything. Note that he plans to take credit for it, so a successful test is essential before he shows it to the boss.
Conversely, Krennic has an over-inflated opinion of his own importance. He thinks that a failed test would diminish him in the eyes of the Emperor, failing to understand that the Emperor just thinks of him as one of Tarkin's little people.
Krennic’s smile faded only a touch. “One had hoped that he and Lord
Vader might have been here for such an occasion.”
Tarkin’s voice was laced with irritation and feigned exasperation.
“And I thought it prudent to save you from any potential
embarrassment.”
My embarrassment, or your own?
Tarkin’s objective was transparent: The man believed (with typical
grandiose certainty) that a demonstration on Jedha would diminish,
rather than enhance, Krennic’s stature. Yet why remained an open
question. Krennic had turned up no evidence of sabotage; nor had his
contacts close to Tarkin revealed anything of use regarding the
governor’s plot. And while Tarkin’s disdain for Krennic was supreme,
he would surely have arranged for the Emperor to bear witness if he
assumed Krennic’s “incompetence” would result in the station’s
failure.
No. The most likely possibility was that Krennic’s precautions against
sabotage or failure had shaken Tarkin’s confidence. The man was now
hedging his bets. If Krennic succeeded in annihilating Jedha, Tarkin
would attempt to take credit in the eyes of the Emperor. If Krennic
failed, all the better.
But Krennic would not fail. The Death Star was ready. Once Jedha was
destroyed, he would receive his private audience with Emperor
Palpatine—and he was confident he could persuade the Emperor that it
was he, not Tarkin, who deserved the accolades.
It even happened to be true.
“Your concern is hardly warranted,” Krennic said. “The finest
scientists and engineers in the Empire have dedicated their lives to
this project. You will not find our faith in them misplaced.”
“If saying it would only make it so,” Tarkin murmured, just loud
enough for the officers to hear him above the din.
Rogue One - Official Novelisation