The difference between the manual method and using Waves Doubler is that the latter lets you modulate the shift amount, and modulate each "doubled" copy separately.
This is supposed to mimic the random interplay of double-tracked vocals. It doesn't. Real vocals are neither random nor sinewave-modulated, and this is no more realistic than randomly moving drum hits in an effort to "humanize" a drum track. That doesn't mean there's no place for the effect; just recognize that it's a gimmick and decide if the performance benefits from it or not. I'd urge you to
not adopt it as part of your default vocal treatment.
What the pitch shift does is generate intermodulation distortion. Normally, IMD is considered a bad thing, but for vocals it can work as long as the effect is kept subtle. IM distortion results in new frequencies being injected into the vocal, comprised of the sum and difference frequencies between the two parts. Perceptually, we hear those new high frequencies as a glossy sheen atop the vocal. As long as we can't zero in on them (because they're low volume and moving around) they aren't perceived as separate tones, much the way reverb modulation prevents us from hearing individual resonances.
I've gotten both good and ugh-puke results from this method. Depends on the vocal, and how natural-sounding you want it to be. If you find that your 1-cent (1 cent = 1% or 1/100th of a semitone) adjustment is too subtle, start raising the pitch amount one cent at a time until you can clearly hear the effect. Then back it off by 1 cent. You may find the sweet spot at 3 cents or even 7.
I would also suggest a band-pass filter on the cloned-and-shifted parts. Most algorithms introduce artifacts that are mostly in the high frequencies, and low frequencies don't benefit from it and only get muddy. Treat your "doubled" tracks' EQ like you would a reverb send. As with a reverb send, it may also help to route your doubled parts to a separate bus so you can easily set their levels relative to the main vocal. Most of the time, you'll want them 3 to 12 dB down.
Have fun!