New Game+ or new character?

New gasme plus just picks you up from where you left off in your previous playthrough in terms of level so you don't hasve to re earn those points. If you start from a completely nwe character then you'll go back to level 1. Personally I don't use the New game plus modes  as I find it makes the game too boring and easy I want to be able to create and make my chjaracter from scratch each time I play. Thankfully I can but it's up to you to decide how you want to approach it.

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Go new character. New Game + has some quirks that need to be worked out. Some quests are a trainwreck because special items remain in inventory, and if you level too far past lvl80, you're in for bulletsponges last I heard.

I say new game+ if for no other reason you maintain your points, gear, credits, upgrades, and anything else you accumulated. Just respec in the medbay and save, so those points will be available for you immediately on Habitat 7. Also, if you like to be perpetually challenged on insanity, it does so. I'm in game+++ right now at level 110 and the encounters are still enjoyable.

I like NG+ because it gives you a more consistent difficulty. If you play from scratch at a higher difficulty, the start of the game is signficantly more difficult than later parts. I don't like that. In a "first" NG+ you usually start at levels between 50 and 60 and you're not likely to get past the point where difficulty rises again after you max out your equipment at level 80 (at least it doesn't appear so after I gained four levels between the start of the game and Voeld, with a completed Havarl, in my current NG+). IMO, the first NG+ after a "from scratch" game gives you the most satisfying experience unless you find Insanity at NG+ too easy.

The only significant disadvantage is that your companions can't improve or gain any more skills after level 53.

I started NG+ at level 43, and I find it great, mostly because the Nomad is upgraded and I did not have to re-level my new guy (I really liked it that you can rebuild the character cosmetically, woot!), and the starting gear was good enough for Normal Difficulty.

I tried that, but couldn't find the special items. What category are they in in the store screen?

I go with NG+ because I don't have to adjust all those details of my character's face again. I just change some make up and hair style.

Also you can and change your abilities any time you want, also you can go reset your skill point.

Quest items are tied to the character and can't be gotten rid of. I'm just about to do the Moshae mission and so far Lexi is the only one bugged for movie night so I'm avoiding any basic open dialogue with her. I guess I'll see what happens. 

What I don't like about NG+ is that it has the same issue that some of the Dragon Age DLC had, once you get past a certain level you pretty much dump your points wherever. Neverwinter Nights addressed this issue with Epic characters, which needed careful thinking about how they were built. My Sentinel knows 4 moves. I could teach him more but that would be tedious.

I liked how ME2 handled New Game + better. You got a shit-ton of minerals, but had to level all over again. I don't mind killing another 10,000 Kett, but don't look forward to having to search for another 10,000 rocks to build a shotgun.

I could use the medbay to re-spec my character, but something about that doesn't feel right.

I started a game+ based on my level 59 Sara (Infiltrator) from the first play-through yesterday. However, I don't see this as a limitation. While my second Sara (actually called her Janine, to have some differentiation between them) is supposed to Biotic, she starts out with the skills of an infiltrator maxed out - that didn't stop me from putting "push" onto the toolbar and investing the skill point from the first level gain in singularity. I think there are even achievements open I can earn this way.

For now my "Janine" has a fourth favourites set which is her own, plus the three favourites set she inherited from Sara: 1) Infiltrator (Overload, Concussive Shot, Stealth), 2) Soldier (Overload, Concussive Shot, Barricade), 3) Engineer (Remnant VI, Assault Turret, Cryo Beam). Weapons I have maxed out are sniper rifle and assault rifle, pistols are 4/5. So, what I need is to invest a little into shotguns if I want to play a front line adept, but the rest goes to biotics and I will rise in the "Scout" profile along the way.

So even if I do something completely different now, I benefit from the previous skills set... Thus I wouldn't know what speaks against "New Game+". Especially considering that I want to play at increased difficulty and suppose that's true for most players who want to play another game. My first play through was "Hard Core" which was hard enough for me on the first play through - I had my share of deaths, even though it got easier towards the end of the game, as my survivability drastically increased. It would feel self-defeating to not use the benefits of my first play through.

And even if you want different skills... there is this "Respec Station" in the Hyperion's med-bay. Not thatI know how it works, but it should enable you to set a completely different skills set.