What is the advantage of P-Intrinsic-N photodiodes with respect to a single intrinsic semiconductor with a bias voltage?
Your question asks about 3 different types of light sensing devices, a photoconductor, a p-i-n photodiode, and a p-n photodiode. There are pros and cons when using each of these

Photoconductor:
This is a piece of semiconductor with ohmic contacts on the ends. When illuminated electron hole pairs are generated and, if a bias is applied to the terminals, contributes to a photocurrent in the device. The photocurrent gain of such a device can be given by:

$$G_a = \frac{(\mu_n+\mu_p)\tau\mathcal{E}}{L}$$

where \$\mu_n\$ and \$\mu_p\$ are the electron and hole mobilities. \$\tau\$ is the carrier lifetime, \$\mathcal{E}\$ is the applied electric field, and \$L\$ is the length of the photoconductor.

As you can see here, you get the best gain with long lifetimes and short photoconductors. However long lifetimes will make your device less responsive, and short photoconductors will make your device less sensitive. So there are tradeoffs.

p-n photodiode:

A p-n photodiode works basically the same way as a photoconductor, except the built in field helps to separate charge carriers. Since nearly 100% of the voltage drop occurs across the depletion region, this is the only part of the p-n photodiode that has a significant response to photons. Therefore, for a high quantum efficiency, you want a big depletion region, to capture as many photons as possible. But this increases the transit time across the depletion region, increasing the response time. So there is a trade off between quantum efficiency and response time.

Because of the built in electric field p-n photodiodes generally have faster response times than photoconductors. Carrier lifetime also doesn't matter as much in p-n photodiodes as it does in photoconductors since the recombination rate in the depletion region is so low it is usually ignored. 

The dark current in a diode is generally going to be lower than in a photoconductor. This could be important when trying to detect small amounts of photons.

p-i-n photodiode:
The p-i-n photodiode takes the benefits of the p-n photodiode and improves the quantum efficiency by adding an intrinsic region in the middle. This region allows you to tune the quantum efficiency and frequency response to fit your needs. Again, a negligible amount of recombination happens in the depletion region, so carrier lifetimes don't matter much.