Okay, here are my thoughts first in short, then in more detail: In sort- you really can't go wrong with either. They are both the best in the business and for a reason- quality, design, and customer service being top reasons.
That said, the long of the story: Lie Nielsen specifically takes Stanley (in most cases) existing tools and improves upon them, keeping most of the existing traits. Veritas, on the other hand, goes a step further and takes the "idea" of the tool, and creates their own version. The big difference in planes is the set screws, Norris style adjusters, different handle style (old version- now offer a more "Stanley-like" new version handle on their custom planes), and a more modern overall look.
I am not sure who said it- Rob Lee or Tom Lie-Nielsen, apologies to both- but one of them said that Lie Nielsen is like classical music, and Veritas is like Jazz. Both are good. Both are technical, and both have their qualities, but one is classical, and one a more modern interpretation.
Henceforth I will use LN to refer to Lie-Nielsen, and LV to refer to Lee Valley/Veritas to save my fingers from typing.
Now, me personally- I have chosen both brands. My first purchase in the high-end tools was a LN 97 1/2 block plane, and wow the difference from my Stanley. Ahh yes, I now saw the reason to pay top $ for tools, and set out to upgrade my entire collection one by one. My choices were based in part on price, in part on quality, and in part on specific features offered by each maker. Here's the list, from memory, many probably excluded due to my mind being a steel trap... that is- nothing gets out. :-)
Lie Nielsen:
- Small scraping plane- because theirs is bronze, and I'll say it once but it applies to all bronze tools- I live in the tropics, bronze doesn't rust. It is also heavier and I like heavier.
- Cabinet scraping plane- because it is a rabbeting design and I wanted to get up to the edge on lapstrakes in boatbuilding, but also in cabinet making to get to edges... which it turns out I never need to do, but great marketing, LN. I kind of wish I had gotten the standard scraping plane.
- #2 and #4 bronze- see note above about bronze. Also I like the Bedrock design better, although this was before Veritas offered customizable planes, which I have yet to try. I still would probably go with LN because of the bronze.
- Small low angle block plane... ummm... I think it's designated the 103, can't remember. Because bronze! Also smaller than the 62 1/2 and easier for doing small wood strips which is what I mostly do, and it fits better in the apron.
- Chisels- full set of bevel and mortice- because I like the interchangeable handles, the quality that I could depend on (Veritas at the time had JUST come out with their chisels, and ultimately I went with LN because of reputation- BOTH BRANDS ARE EXCELLENT- just my choice) LOVE the mortice chisels especially- hefty, mean business. I do wish LN offered something larger in their bevel edge chisels. I would like to have a wider chisel for tasks like defining the edge of hinge mortices. Ultimately I will probably get a set of LV butt chisels just to get that wider blade.
- Chisels- fishtail and skewed- Does LV make a fishtail? I don't think they do, but anyway I had LN already, and also I like that I can get the long handles for the fishtails and skews. I think LV does sell skewed chisels, but I am very happy with LN, and went with the brand and, yes- matchy matchy won. I hate to admit, but sometimes if all else similar, you go with the ones that match what you already have. Don't tell anyone. I hate matchy matchy people. I do love the hornbeam handles as well.
- Screwdriver set- because I hate when the screw heads strip from using a just slightly too small screwdriver. I love these things.
- Tenon saw- I believe it's 16", and hands-down would prefer LN to LV. Brass back, flamed maple handle. Bliss. I wanted a REALLY good tenon saw, and this is it.
- Scrapers- because I was making an order and had some extra $, but really a scraper is a scraper (not a scraping plane, but just a scraper blade that you hold in your hand). That said, I think my curved scraper was from LV. Not sure.
Veritas:
- Low angle Jack, Smoother, Jointer- Because here is where if you get one brand, you need to stick with it so you can use one set of blades. I have tons of blades including three 25 degree, two 38 degree, I believe I have one at 45 or some higher angle, and the toothed blade for figured woods, which I use all the time. The main reason for going Veritas: The fence for the Jointer, the japanning on the sides (tropics-rust- same as bronze reasoning) and the price was a bit less for what I saw as the same quality. I love these, and since getting the jointer, I don't think I have once used my Stanley No. 7 or No 6. Same goes, in fact, for the No. 5 versus the bevel up LV. I do use other smothers, but since I currently don't have a high-angle frog for my LN #4 (probably my favorite smoother), I use this smoother for fancy grain.
- Shooting Plane- because it's a work of art, and also in my opinion it's the best shooting plane out there. It is heavy,has an adjustable handle, skewed blade, and set screws. It's just pretty to look at. Sometimes I take it out just to look at it.
- Set of rabbeting block planes- my favorite planes. I use them all the time. I love, love, LOVE these!!! They were cheaper than the LN, so even though I would love to have the bronze LN, I went with Veritas. They also have the set screws, and on a rabetting plane that is important, because it is hard to get the blade set just right without those. (like when you take it out to sharpen it and put it back in)
- Large shoulder plane- Cheaper than LN, high quality, and I liked the bubinga knobs versus the LN handle. I also like the set screws. It turns out to be rather cumbersome, and I will soon get a medium.
- Bullnose shoulder plane- Because really I wanted a chisel plane, but this gives both, as the bullnose is removeable. Also as mentioned above, the large shoulder plane was more cumbersome than expected, and I really wanted a medium, but I got this instead for versatility.
- Router plane- Friggin love this thing, and the whole kit with the string inlay tools is why I went with LV over LN. No regrets whatsoever.
- Beading tool- because of price mainly, and I hate to admit this- but because it matches the set of spokeshaves! Yes- matchy matchy, I hate myself. Also I liked the fence and thought it looked easier to register. I don't, however, like the push versus pull stroke.
- Full set of spokeshaves- Because (a.) I love bubinga handles (b.) Price (c.) Joe O'blenis (Joe O' Paddles) raved about his, and I respect his opinion, and (d) Really just as good as the LN, although I sure do like the bronze LN. Just can't afford.
- Dovetail saw, crosscut carcass saw- because of price. I much prefer LN. The spine on the LV does flex. I use the LV saws as my go-to for basic projects, and I have other high-end saws that are for special projects. Won't mention those here to keep it just LN versus LV I do highly recommend LV saws and use them often, but my preference if money were no object would be LN.
- Honing jig with angle guide- because I love theirs. I find nobody has really made one that covers all bases, but LV is the most well-rounded that I personally have used, and I have used many.
- Marking gauge- love the round marking gauges with the wheels. I think LN also sells similar, but LV won me on free shipping if I remember correctly.
- Striking knife- I think I got this one on eBay and it was cheap, so I bought it. Like it much better than the ceramic one I have, but not as much as the Japanese one.
I am sure there are others that I am missing here, but basically each tool I chose one or the other mainly because of certain features, or if all else relatively equal, then I chose on price. You can go all LN, or all LV, or a mix of the two, and either way you still have great tools.