Here is a link to a chart with some differences between them:
http://www.yamaha.com/yec/compare/Detail.html?compitem1=&compitem2=567570&compitem3=567571&CTID=5000300&VNM=LIVE&B_compare.x=31&B_compare.y=10&comp_items=567570&comp_items=567571
There is more different than just the output power and the number of inputs. For example, the RX-V863 has an HD radio tuner and can do video up-conversion.
You might also want to download the manuals and read them; you will have to register with Yamaha to do this, but it is free.
The different stories you hear about the output power for each is a result of the fact that an amplifier can be measured in more than one way. One should look at continuous RMS output into a specific impedance at a specific amount of distortion over specific frequencies. Changing the impedance one is considering, or the level of distortion, or the frequencies, will enable one to have different numbers claimed. So, one amplifier can be said to put out different numbers of watts, depending upon those variables (not to mention "peak" power output). And this also means that when comparing two amplifiers, all of those must be the same, or nearly the same, to be a fair comparison.
As for the specific models in question, the amount of power difference is fairly insignificant. The RX-V863 is rated at 105 watts RMS from 20 Hz to 20 kHz @ 0.06% THD @ 8 Ω, and the RX-V663 is rated at 95 watts RMS from 20 Hz to 20 kHz @ 0.06% THD @ 8 Ω. Since everything else is the same, we can directly compare the number of watts claimed here. Assuming that they are accurately rated, this difference will probably never even be noticed. The reason for this is because in order to get just a 3 dB increase in volume, one must double the power. So, a difference of 10 watts, if we were talking about a 10 watt and a 20 watt amplifier, would be 3 dB. But to get that same 3 dB increase over 95 watts would require 190 watts. 10 watts at the level of power you are dealing with is trivial. If power is the only issue, you should probably go with the RX-V663. But for a $100 difference, I would go with the RX-V863 for its added features, such as HD radio and an extra HDMI input. It also does video up-conversion, though I do not know if it does it well enough to matter. (With video up-conversion, this can be done in a source unit, the receiver, a separate video processor, or one's HDTV; it does not matter which one does it, but you want it done well [if you watch any non-HD sources on an HDTV], and not all up-converters are equal.) You will want to read each manual if you want to find all of the differences between them.
As for your girlfriend's advice, she is pointing out the least expensive model in Yamaha's lineup that decodes the new audio formats on Blu-Ray (though they have newer models now on their web site). If you don't need the extra features of the RX-V863, she is giving you excellent advice. If you do need them, then when getting advice from her, you should tell her more about what it is that you need. If all you told her was that you want a new surround receiver, and expressed nothing more, her advice is exactly what I would have told you to get. The RX-V663 is a great receiver for the money. If you were paying retail, I think the price difference would be such that the RX-V863 is not nearly as good of a value, as there is a $450 retail price difference between them. It isn't that much better, in my opinion.