What tonal differences can I expect between a 1 x 12 cabinet and 2 x 12 cabinet when using the same amp and same type of speaker?
Does changing how the 2 x 12 cab is wired (series vs parallel) affect the sound produced?
It's all about headroom
2x12 will get you a lot more headroom, 4x12 even more.
It's not about volume, if you have a 1x12 that is 100W and a 2x12 that totals 100W, for the same settings in the amp, you will have the same volume, but in reality the 2x12 will seem louder and the sound will be a lot less muddy with way less tweaking of the EQ.
That's what we mean when we say headroom. We are basically speaking of the amount of sound that is still available after you've tweaked your setup and are comfortable with the sound.
A 2x12 gives you a lot more room to work with.
About the series vs parallel, that will depend mostly on the amp. As you may or may not know, the impedance will change with changing the speaker configuration. Some amplifiers change more or less the tone according to your selected output, but most of the times it is about headroom too. I don't have facts to support this, but appears to me from my personal experience that lower impedances give you more headroom. Most of the time, people do the wiring on a cab according to what suits them best. Say you have 2 8 ohm outs on your amp and want to connect a 2x12 and possibly later, you will want another 2x12, you should wire it to make 8 ohm so you can expand when you have the other cabinet. 8 ohm is what usually amplifiers have multiple outs for, there's usually only one 16 ohm and/or one 4 ohm. I'd recommend you buy either two 4 ohm speakers and wire them in series or buy 2 16 ohm and wire them in parallel to make the 8 ohm.
You can also look around for a schematic to change the impedance on your cab to switch between different setups. Say you buy 2 8 ohm speakers, you can put them in series (16 ohm) or parallel (4 ohm) and then you can try for yourself and listen for yourself.
REMEMBER TO NEVER SWITCH WHILE THE AMP IN CONNECTED TO THE CAB IF YOU DECIDE TO DO THIS!!!
Unless you have an amp that adjusts automatically according to the speakers connected, which i've seen. These usually have only one output and it says 4~16 ohm
Antero's answer has the wiring/impedance aspect well covered, so I'll leave that alone…
but tonally, what you introduce in a 4x12 or even 2x12 over the single 12 equivalent is phase-cancellation; which can emphasise or reduce frequencies in a comb-filter pattern.
Not that this is bad, in a guitar rig - it's actually a significant part of the characteristic sound. The classic 'Marshall 4x12' of the 60's, with its characteristic 2 vertical, 2 sloped design has influenced the way we think a 4x12 should sound like to this day.
The sloped front 4x12 does sound different to the 'flat front' alternative, even though I think the original intention was just that the top pair pointed a bit more towards the player, rather than being a conscious sonic design decision.
My acoustic theory is less than perfect, so rather than me attempt to fumble a technical explanation of how this happens, I'd say go listen to some, side by side; you'll quickly hear what I mean.
Wow great stuff here! I am a sound system designer and acoustician with over 30 years experience, as well as a guitar player. The idea that whether the cabinet has 1 12 or 4 12s, if connected to one head, they will output the same acoustic energy is correct. You will not move more air, but the air you move will be much cleaner and sonically that clarity translates to a more "powerful" experience for the listener. If you actually want more acoustic power you have to increase your power amplifier output o add a second amp. The reason most players in a typical bar do well with a 60W to 100W setup is that going to 120w or 150w does not increase your acoustic output significantly.
If you have one setup that can produce 100dBA at the front of the audience (just for arguments sake) adding a second identical setup paralleled to the first will increase the level 3dB. This difference is barely audible to the average listener.
The phase cancellation or combing effect is also correct. This is simple to map out with two sources (2 12s) and gets more complicated and different when there are 4 12s, especially if they are not in the same plane.
In the end, this is rock and roll. those that try too hard to quantify what is going on here are missing the point. Take CBS buying Fender. The turned around and created the cleanest transistor amplifiers ever known and were so happy they finally cleaned up this goofy business that Leo had created. It was one of the biggest corporate failures ever known. I believe they talk about it economics classes. Nobody wants a perfect guitar amp. Perfect is what you hear and love.
A 2x12 cab will be moving twice as much air as a 1x12 cab. A 4x12 cab will move 4 times as much ! The more speakers, the better the sound, generally. Not necessarily louder, but that will depend on impedance match. Be careful that the polarity of a pair of speakers is correct. If they're out of phase, they will to a degree cancel each other soundwise. As in the bass response for a start will be reduced.Series or parallel won't change the sound, but will give an impedance mismatch with the amp, although a lot of amps are forgiving.Also bear in mind that an open back cab. will sound quite different from a closed cab.