First of all, the DV 336se is meant to drive high impedance headphones (let's say >200ohm), and it won't sound good driving anything with lower impedance, so you could only drive the DT770 and HD6XX.
Second, the difference is very subjective: some people might prefer the more detailed & analytical sound of the LD, some other might enjoy the warmth & soundstage of the DV. Both have more than enought power to drive high impedance headphones like the DT770 or HD600/650/6XX, specially the DV: LD is meant to drive lower impedances, like planar magnetics, but is capable of driving higher impedances pretty good, while DV won't drive any low impedance headphones without sounding like crap.
Third, my comparison is not "fair", because I've rolled tubes & opamps on the LD until finding a good setup for both my HD6XX and HE-4XX (ask for suggestions if you wanna know, those sell really cheap and difference is 100% worth), while my DV has stock chinese crappy tubes (I wanted to buy better tubes, but couldn't afford it until a few days ago).
Now, from my experience, I prefer the DV for the HD6XX but not by a huge margin. It sounds more pleasant from the DV, indeed, and has way better bass (and a bit more sub-bass too). Soundstage is bigger too, and DV adds more "tube sound" (aka harmonic distortion) and warmth than LD. But on the other hand LD sounds more "accurate", has more clarity & detail and it's easier to tell minor sonic differences between records. I'd say LD is better for analytical listening, and DV better for relaxed listening. And since HD6XX are not too analytical per se, both bring something to the table: a bit more detail to compensate for the natural character of the headphones, or even more warmth and forgiveness for the ultimate relaxed listening experience.
It also depends a lot on the music you listen: for bassy genres like EDM, hip hop or anything like, DV is clearly the way to go, giving the HD6XX a much more impactful, beefy, textured bass (but controlled and non intrusive). On the other hand, for classical music LD might be more revealing and richer in detail, allowing you to hear nuances that DV will probably not.
I guess better tubes on DV will improve the sound even further: according to my readings there's a huge difference after upgrading tubes, way more than on hybrid amps like the LD I+, and you can also customize the sound according to your preferences by using different models.
To sum up, I'd say LD I+ stock vs DV stock is a unfair comparison: LD can be bought for <$100 while DV drops for ~$200. And LD with better tubes and opamp is still cheaper than stock DV (maybe you can spend ~$130-150 for the full package), and tubes for the DV are way more expensive too, so they play on different leagues. Is it worth the upgrade? It depends on a lot of things, it's hard to tell. What I can tell you, is that if you plan to drive several high impedance headphones and you enjoy the "tube sound", DV is probably the best price-performance "budget-ish" amp right now. But get your wallet ready, because stock tubes must be replaced in order to enjoy the "full experience", and replacements are not cheap: after a lot of searching on eBay, I found power + driver tube for ~$80 total, inc. shipping, because I'm splitting shipping costs with a friend; but those are not the best tubes you can find, by far.