I don't think it's explicitly mentioned in any source why Palpatine decided not to publicly connect Vader to Anakin. However, it's not difficult to make an educated guess.
I've explained in another answer why Palpatine hid his Sith identity from the public: Palpatine's story that the Jedi had attempted to take over the Republic depended greatly on the public perception that Palpatine was an old man viciously attacked and deformed by the Jedi attempting to seize power. Palpatine's image as a kindly old politician would conflict with his image as a powerful but evil Sith Lord. Furthermore, Palpatine was most effective as a politician rather than an overt combatant anyway. There was no benefit to revealing his Sith identity but there was a huge disadvantage.
The same is not true of Anakin/Vader, though. Anakin was a well known and popular hero of the Republic during the Clone Wars. The Empire even preserved Anakin's image after his "death", claiming that Anakin had died defending Jedi children against "Vader's" 501st in the Jedi Temple. On the other hand, Vader is an instrument of terror:
It was that genuflecting obedience, the steadfast devotion to execute whatever task the Emperor assigned, that had given rise to so many rumors about Vader: that he was a counterpart to the Confederacy’s General Grievous the Emperor had been holding in reserve; that he was an augmented human or near-human who had been trained or had trained himself in the ancient dark arts of the Sith; that he was nothing more than a monster fashioned in some clandestine laboratory. Many believed that the Emperor’s willingness to grant so much authority to such a being heralded the shape of things to come, for it was beyond dispute that Vader was the Empire’s first terror weapon.
- Tarkin, p. 73
Just as Palpatine's Sith identity conflicted with his public image as a kindly old politician, so too would Anakin's heroic Clone Wars image conflict with Vader's image as an instrument of terror. Thus, Anakin had to remain publicly separate from Vader.
Furthermore, tying Anakin to Vader as you suggest would invite questions about what happened to Palpatine when the Jedi "attempted to take over", and why Anakin/Vader did what he did. But Palpatine needed to avoid any questions about that incident to make sure no one discovered what really happened:
There were many stories about what had occurred that day in the chancellor’s office. The official explanation was that members of the Jedi Order had turned up to arrest Supreme Chancellor Palpatine, and a ferocious duel had ensued. The matter of precisely how the Jedi had been killed or the Emperor’s face deformed had never been settled to everyone’s satisfaction, and so Tarkin had his private thoughts about the Emperor, as well. That he and Vader were kindred spirits suggested that both of them might be Sith.
- Tarkin, p. 77
As for your question about how Anakin/Vader would have served Palpatine in his original form (no transformation into the suit): while it's always difficult to guess what would have happened in a hypothetical, I'd imagine Vader would have been used less as a terror weapon and more as a righteous former Jedi still attempting to restore order after the Republic was torn apart by the Clone Wars and the Jedi. The Empire's terror weapons would have instead been the stormtroopers, Death Star(s), etc.