What's the difference between dual antenna and triple antenna for a router?

It depends on the radio(s) those antennas are connected to.

Let's say you're comparing two 802.11n APs. One has two antennas and the other has three. Well 802.11n's big speed boost over 802.11a and 802.11g is that it can uses MIMO -- ganging together multiple radio chains to boost performance. It takes two radio chains to reach signaling rates up to 300 megabits per second, and with a third radio chain you can reach signaling rates up to 450 megabits per second. So if it turns out that both APs have one antenna per radio chain, it might be and indication that the three antenna unit can go 50% faster than the two antenna unit.

But you can't just count antennas to know how many radio chains they have, you have to confirm it by reading the technical specifications of the device.

If the two or three antennas are all hooked up to a single radio, like in the case of a single-band 802.11g AP, then the third one probably doesn't matter much. With a single radio, it makes sense to have two antennas -- main and auxiliary -- for the sake of antenna diversity. Sometimes the main antenna is in a better place to receive a given signal, and sometimes the aux receives the signal better. A third antenna in this case is just another aux, and it's probably not going to help much given that they're all within a few centimeters of each other. But then again, the antenna designer may have planned out the coverage patterns of those three antennas to complement each other well, so it's possible it can make a difference. But you wouldn't really know unless you had an antenna engineer carefully test it.

Overall, you really can't judge what a wireless router can do in terms of rate-vs-range just by counting antennas.