Hi Glacies-
Lots of thoughts on this. I'll try to organize them accordingly. I thought your vocals sounded pretty good, so nice work on those.
First off, I would try using a backing track like this one: http://download.guitarbackingtrack.c...nite%20(3).mp3
A big part of why the distorted part sounds so weird (outside of transitions) is that you just can't get that much of a contrast in dynamics from a guitar alone. Drums and bass are much more important to making a band sound like a band than a guitar is, though guitar and vocals are usually the most noticeable components. So a lot of that intensity will come through the drums.
I would also try using a lot less gain. In general, reverb and gain are way more present on recordings than they are live, so it ends up sounding better to turn both of those way down. I noticed your vocals were pretty reverb-heavy. That may be due to your recording setup, but see if you can tame that at all with some foam or other isolation. Ironically, I thought your arpeggiated rhythm guitar could have used just a little reverb to help it sit a little better. Right now it sounds a little dry.
I also think that playing the full arpeggios on the rhythm would fill it out and make it sound better. I don't know this song note-for-note, but I think they play more notes on the arpeggiated part, like maybe 4 notes per chord on the ascending part at the end of the phrase instead of 3. The gaps in there now make it sound a little awkward.
You can really help to fill out the sound a lot by doubling (or tripling/quadrupling) your rhythm guitar and panning it slightly left or right, and compression can help get a more intense sound. I have not done a lot with that, but it's kind of conventional wisdom WRT recorded guitars. I have done the doubled thing and it does help a lot. I would think you would want to do that with the chorus for sure.
You are correct that the transitions could stand to be cleaned up. It sounds like you may be having trouble monitoring yourself and playing at the same time. What I have found to work is that you listen to the mix on headphones and play back along with the track, then just chop out or mute the sections you don't want to be playing on (e.g. the verse for the overdriven rhythm guitar). The tricky part is routing everything properly through your computer and recording interface so that you can hear the playback while also getting your signal into the computer. What does your setup look like right now?
If you have a good recording/monitoring setup, multi-tracking should not be too difficult and I agree it is probably the best way to approach this sort of thing. If you multi-track you have control over the gain for all of the parts so you can turn up just the guitar solo, etc. and EQ as needed.
I think using a backing track like the one I linked to would help you know where you are in the song, stay on rhythm, and ensure you nail the transitions. I don't doubt that you sound pretty good in a live setting, but recording is a very different animal and is pretty tricky at first. This is probably what you actually do sound like, but the little nuances and imperfections that your ear is drawn to on a recording don't matter nearly as much live, so it's annoying at first to try to get a decent recording down.
One last thing is that on your software/DAW, there is usually a time-shift thing if you need to shift your track over at all due to latency. So as long as you are in rhythm, you can usually correct after the fact to sync up one part with the rest of the tracks. I think it's good that you are getting into this; it's an important part that I have avoided for years but am now forced to become more familiar with.
I see that this ended up not being as organized as I would have hoped, but I think I got all of my thoughts in there, so hopefully it was helpful. Thanks for posting this!