If you are talking about a 802.11n (draft) router then the multiple antenna are for the MIMO functionallity, that improves reception in multipath environments.
As for 802.11a,b,g routers, The second antenna is for 'diversity' The radio picks the antenna with the best signal and uses that antenna. Both antennas are not used at the same time. Some routers with one antenna have a second diversity antenna internal to the box.
A router with a replaceable antenna would be something to consider, most stock antenna are very poor.
Thr patter is different between the 2 setups. A single antenna is omni directional. Where the dual antennas is like a figure 8 with the base in the middle. Similar to a beam antenna, but shooting backwards and forward with some side coverage.
The most important factor will be the dbi gain of the antennas. Higher the gain better the sensitivity which improves range and signal strength.
Before doing anything with wireless.... ask yourself
What range do you need to cover?
Do you have any odd elements in your raido enviroment such as a stucco building or metal roof/beams?
I'm guessing that this is a "normal" home setup and that you want maximum range out of a single wireless router.
If this is the case then the antenna is not the biggest part of the picture. Do your research and pick a device that suits you best.
If you are using any speed bost or similar technology make certian your device (router or WAP) and the NIC in your computer support the same thing and you will get the most out of the system as a whole.
I would take jjw's adivce though and get one with replacable antennas. I am a huge fan of the D-Link products and use them (albeit with a celing mount omni) in my buisness instilations.
The configuration of my house is quite simple - a single floor apartment of two and a half rooms in size with very thin walls.
Tests I've conducted in the past guarantee that even the simplest of routers will do the work to my satisfactory.
My intent was to determine whether I'll be able to recieve anything extra from the second antenna, such as better reception outside of the apartment or even at street level, if possible (I'm at the fourth story).
So far I understand from your answers that the second antenna changes very little, which, if correct, leads me to wonder why would the manufacturers decide to add it to the device at all.
There is a difference in coverage. The dual actually have a longer range than the single. But the direction is front to back. The single gives a more uniform coverage. Most with dual allow you to turn off either side if you want. The main thing that governs range is sensitivity as in dBi. The dual antennas have the effectiveness of a high gain.