I am currently an E-Nano owner/user, though I literally just ordered an Underdog a few minutes ago.
As has been mentioned, E-Nano runs on 120V
AC power, the Underdog on 12V
DC power. For me, and I fully admit this is me being neurotic/OCD, but powering something from direct AC power I personally find a bit annoying. For one, typical residential AC (also called "mains power") isn't consistent. That goes for region to region, house to house, outlet to outlet within the same house, and even the same outlet at different times. In practice, does it really matter? No. But combine that with the imprecise dimmer that is included, and my "6.5" is almost certainly not the same as your "6.5". Even my own 6.5 at noon might be different from 6.5 at 9:00pm on the same outlet...
I have yet to see if the UD power supply is regulated or not. If it's
not regulated, then it will suffer the same inconsistency problems as the E-Nano. Likewise, I suspect the cheap
dimmer that can be bought with the UD will suffer similar precision problems as the Nano's dimmer.
However, as far as I can tell, DC power is much easier to precisely regulate. Hence, the variable voltage power supply (VVPS) you hear everyone talking about in UD context. The VVPS is very different than the dimmer (I often see them erroneously conflated), in that it offers precise
regulated voltage control. So when you dial in 11.9V on your VVPS, the circuitry takes care of AC-level fluctuations. On the same VVPS, my 11.9V is the same as your 11.9V. Even without going the VVPS route, regulated 12V power supplies (even those of reasonable quality) can be had for little money ($10-20).
Just to be clear: do you really
need precise voltage control of your log vape? Probably not, being in the right ballbark is likely good enough for most people. But if you have OCD tendencies like me, it might bug you.
I haven't seen a way to get precise AC voltage control, at least not without spending a lot of money. One option would be to use a variac variable transformer; some of these actually have gauges on them that tell you what the output voltage is. The cheapest suitable variac I've found is the PHC Enterprise SC-3M. It's $65 and also big and heavy. Some of the nicer UPSes (uninterruptable power supplies) will do AC conditioning, and output a perfect 120 VAC sinewave. But these are bulky and expensive. There are also line conditioners for audio equipment. And also specialized "bench" power supplies for lab use.
Next consideration would be cord size. The Nano's cord is a sturdy 18 AWG. I spent a fair amount of time searching the web for smaller cables. When you search for cables specifically branded for AC/mains voltage, it seems smaller than 18 AWG is a rarity. Of course you can buy bulk cable in any size and make your own. As far as I can tell, it's
current (amps) that matter, rather than potential (voltage) when choosing wire thickness. And for the tiny amount of current the Nano draws, I believe 18 AWG is total overkill. Even a noticeably smaller 22 AWG I think should still be overbuilt for this purpose. That said, I did manage to find a 22 AWG power cable on ebay; it's shipping from China so it will take a while before I can put it to the test (and I'm banking on the listing being accurate, which, for a $1.50 item, might be a stretch!).
Again, it's a matter of personal preference, but that thick cable on the Nano really annoys me. But there's a huge number of people who aren't bothered by it. It's not a deal-breaker for me, but definitely an annoyance.
The upside to AC, though, is that cable length matters less (and this is one of the main reasons why the world standardized on AC for power distribution). You've probably read about people cooling down their Underdogs a bit by using a longer cable. That trick won't apply to the Nano.
As has been mentioned, if you want to make your log vape somewhat portable, batteries are inherently DC, so battery packs are generally DC. There
are portable "AC" battery packs available. But they are far less widespread than 12 VDC battery packs, which are effectively ubiquitous. That means you have more options with DC battery packs and generally cheaper prices. Also note the conversion from DC to AC is not 100% efficient, so you'll waste some battery power in the conversion process.
Other small differences: if you look at pictures, you can see the Nano has a metal "barrel" that hugs up against the wood. I don't believe the Underdog has this. There was some speculation in the E-Nano thread that this barrel helps the Nano preheat air before it goes through the actual heater core. Seems reasonable to me.
When placing my UD order, I noticed the UD stems don't have the nice octagonal silicon sleeve that the E-Nano stems do. These are nice for keeping your fingers from getting burnt, and, perhaps more importantly, prevent the stems from rolling. I'm hoping I can use the Nano silicone stem guards on the UD tubes; but I know UD and E-Nano stems are different sizes, so that might not work...
Just to be clear, should I be giving the wrong impression, I really do like the Nano. I did not when I first bought it. I was coming from an Arizer Air, which definitely wins in the
convenience category. But now that I've had more time with the Nano, I've come to appreciate all the areas where I think it dominates the Air: simplicity in terms of design/construction - my Air has had zero problems, but it's inherently more complicated than a log vape, so I believe it has a greater potential for long-term reliability issues; vapor quality on the Nano is heads and shoulders above the Air - in terms of flavor, potency, draw resistance - two (maybe three) solid rips off the Nano gets me to the same place as a whole 10 minute session on the Air.
Those are my thoughts/observations thus far. I'm sure I'll have more to say when I receive my dog!
Edit: Another small thing the Nano has going for it: the infinitely versatile
E-Pick tool that comes with it. Granted, there are 100 common household items that will have the same utility... but, the E-Pick is just so perfect. If you read enough of the E-Nano thread here, you'll see how it is used to to measure the "standard" screen depth and also product depth (of course you can use whatever depth you want, but these depths seem to work for a lot of folks). Plus it's nice for stirring and lightly tamping product, or even as a makeshift grinder if you like to throw whole nugs in your tube.