I go back and forth on whether the purpose of the diary Horcrux was to reincarnate Tom Riddle (and note the difference between reincarnate -- to give another body to; to incarnate anew -- and to protect a person, Voldemort, from death). The purpose of a Horcrux is not to create a new individual, but to protect its creator from death.
That said, the diary Horcrux is the only of Voldemort's Horcruxes to serve two purposes:
"What intrigued and alarmed me most was that that diary had been
intended as a weapon as much as a safeguard," [Dumbledore]
"I still don't understand," said Harry.
"Well, it worked as a Horcrux is supposed to work -- in other words,
the fragment of soul concealed inside it was kept safe and had
undoubtedly played its part in preventing the death of its owner. But
there could be no doubt that Riddle really wanted that diary read,
wanted the piece of his soul to inhabit or possess somebody else, so
that Slytherin's monster would be unleashed again."
"Well, he didn't want his hard work to be wasted," said Harry. "He
wanted people to know he was Slytherin's heir, because he couldn't
take credit at the time." (Half-Blood Prince -- Chapter 23)
However, canon is a little inconsistent. In Chamber of Secrets, Tom Riddle clearly becomes corporeal in some form, feeding off a "strong diet of [Ginny's] deepest fears and darkest secrets." He grew powerful enough to feed a bit of his soul back into Ginny. Tom Riddle says about Ginny, "She put too much into the diary, into me. Enough to let me leave its pages at last . . . " (Chamber of Secrets -- Chapter 17). While Half-Blood Prince references the diary as a weapon to possess someone else to work on Tom Riddle's behalf, Chamber of Secrets shows that he was able to return to some kind of form.
Tom Riddle is aware that he is also Voldemort and seems to possess Voldemort's future knowledge and memories:
I AM LORD VOLDEMORT
"Twice -- in your past, in my future -- we have met. And twice
I failed to kill you. How did you survive?" (Chamber of Secrets --
Specifically, if Tom Riddle had indeed been reincarnated, he would have taken the form of Tom Riddle, who at sixteen years old was not yet Lord Voldemort, although he had decided on the moniker for himself while still a student at Hogwarts. Therefore, Tom Riddle would have co-existed with Voldemort as separate entities of the same life force; it would not be Voldemort clones running around.
Would Tom Riddle and Voldemort have competed against one another? Not in the sense of, "Hey, let's grab a pick-up game of Quidditch!" Dumbledore says:
"Voldemort likes to operate alone, remember. I believe he would have
found the thought of being dependent, even on the Elixir [of Life]
intolerable [ . . . ] Therefore, I am convinced, he intended to
continue to rely on his Horcruxes: He would need nothing more, if only
he could regain a human form. He was already immortal, you see . . .
or as close to immortal as any man can be." (Half-Blood Prince --
I think Tom Riddle would not be a match for Voldemort; at sixteen, Tom Riddle simply had not developed his magical abilities to Voldemort's level. My guess is that the magically superior Voldemort would have used and exploited the reincarnated Tom Riddle for every possible purpose (whatever purposes those might have been) and then killed him. I postulate that the reincarnated Tom Riddle, born of the diary Horcrux, would not himself have been a Horcrux, just as Ginny Weasley was not a Horcrux even though Tom Riddle's soul occupied and possessed her -- it was the diary itself that was the Horcrux. Therefore, Voldemort could have disposed of the reincarnate Tom Riddle without harming himself.
There ain't room in this world for the both of them ;)