It's all personal preference. If you want to stick close to stock,
HAPP Competitions or
IL Eurosticks are the way to go. Most people seem to prefer the IL over the HAPP, from what I've heard. However, I have two Neo control panels, and both had
HAPP Ultimates on them when I got them, so I'm wondering if that's what SNK used originally. General consensus seems to be that Ultimates are far from Ultimate...
If you want to go the Japanese route, then you either want the
Sanwa JLW Bat Top or the
Sanwa JLF. The JLF is the preferred Sanwa stick, but the mounting hardware is different from a standard US cabinet like a Neo, though it's not difficult to modify to fit. Many pros prefer these, though I think the casual player is probably more indifferent. The JLW's mounting is US styled, meaning it will drop in just like the Competition or the Eurostick, but retains the Sanwa 'clicky' feel. Most people don't seem to like the JLW as much as the JLF.
Personally, I ordered one of each (sans the Competition) a few months ago and have been spending time with each one to try and choose which model I want to go with long term. Here are some of my opinions:
HAPP Ultimate: I don't much care for them, myself as I find myself hitting diagonals too easily and end up crouching or something when I don't want to. This is probably just me, but I've been unable to adjust my playing style enough to counter it. After 5 years of trying to work around that, I decided to give other sticks a shot.
HAPP Competition: This is or is close to what most people probably grew up on here in the States. As such, the height and feel of the stick is about right and will be very familiar to most players. I don't actually have one of these, believe it or not, to accurately compare to the others here.
IL Eurostick: Basically the same at the Competition in all aspects. They feel damn near the same, from what I've read. Compared to the Ultimate, these are a little taller, which I personally like. The IL feels a little higher quality vs any HAPP I've ever used and tends to be preferred by most people who have tried them (when compared to the Competition). I have not used mine enough yet to know if I have the same issue with it and diagonals yet, but I do tend to like the feel of it better when compared to the Ultimate.
Sanwa JLW: As I mentioned above, if you like the Japanese style sticks or just like the louder click they make, but want a bat top (most are ball tops) and one that will drop in your cab with no modifications, then this is a good choice. The stick is quite a bit shorter than either of the above sticks, which is REALLY noticeable at first. The feel of the bat itself seems a little 'cheap' to me, especially when compared to the IL, though I do like the metal shaft and how solid the whole thing feels. It also has an easy way to change between 8 way and 4 way, which is a nice touch, but largely unnecessary on the Neo. However, Neo Mr Do! with a 4-way stick is A LOT better.
Sanwa JLF: Again, if you really want a good Japanese stick, this is the one to go with. Standard, it's a ball top, but you can buy a bat top for it if you want (which is what I did). As I mentioned, the mounting plate is different (made to fit candy cabs), but you can make your own or even buy a mounting plate for the JLW and attach it to the JLF so that it fits your Neo - pretty easy. It comes with a square restrictor insert, which is sort of like what the HAPP and IL have/feel like, but I prefer to use one like
this, which helps with my apparent diagonal issues

The square one it comes with can be rotated to switch to 4-way mode, much like the JLW. Too bad Paradise Arcade seems to not carry it anymore/right now (
link), because they let you buy it piece by piece so you could get exactly what you wanted without ending up with extra parts.
In the end, I'm probably going to go with the JLF + bat top, but I need to find another one plus another JLW mounting plate to use with it. Right now I have an Ultimate and a Eurostick in my cab.