I have both these models, purchased new within the last 12 months. For context, these are my first Fenders after playing exclusively Stingrays for the last ten years.
I'm not sure of the OP's location but here in Australia there was a big price difference with the Am Pro II being much more expensive.
Here are some general observations and comparisons.
IMO they are both very good basses for mass produced instruments. Although I have owned better I have no complaints with the overall build quality of either bass. The neck pocket joint is definitely tighter/better on my MIJ H II than on my AM Pro II but both are acceptable. This might be down to the variance between individual instruments.
The AM Pro II had a really weak jack socket which needed replacement almost immediately.
Both basses were in a terrible state of setup from the shop and both needed a fair bit of work to get to what I'd consider playable. With that out of the way, they both setup pretty well (although not as good as my Stingrays, sorry
).
The AM Pro II has a hex key truss adjustment at the heel, and comes with a toolset which includes a T Wrench which allows adjustment without removing the neck. The JP H II adjusts at the head, but didn't come with any tools.
We've had some seasonal changes recently where I live and the AM Pro II neck was more stable than the MIJ H II which needed adjustment when the AM Pro II did not.
The necks on both are very good. Necks are very personal things. I prefer the MIJ H II but that's just me and I would not argue with anyone who preferred the AM Pro II. They are both really good. Frets are very good on both, splitting hairs but again I prefer the MIJ H II, but this is a personal thing and it's line ball.
The AMP Pro II has the contoured neck heel thing for access to the higher frets, the MIJ H II has the standard four bolt positions. Either is fine by me but might be important to others.
The pots feel similar on each. The tuners on the AM Pro II feel better, but both work perfectly fine.
I forget what wood the bodies are made of but the AM Pro II is *significantly* heavier. I can't overstate this, the difference is huge and noticeable. They both hang on the strap nicely and as a P Bass should, but for anyone who is sensitive to heavy basses, I strongly urge you to pick up a AM Pro II before you decide to buy one. It is a LUMP of a thing compared to the MIJ H II. I'm fine with heavy basses but if I wasn't then the MIJ H II would be better.
Both sound great to me. The AM Pro II has the greasebucket circuit. I am still getting used to this and I'm not sure where I stand on it. For sure it rewards a careful hand on the tone pot to get the best from it. If you just want a plug and play, set and forget classic P bass tone then the MIJ H II is easier to get there out of the box, but they will both do it of course.
I've gigged both basses extensively and been very happy with each. If I could only keep one I can't honestly say which one it would be. Perhaps maybe *just* the AM Pro II because of the neck stability and the fact that it maintains a setup a little better. If I could only afford one and money was any sort of an issue, then the MIJ H II all day every day.
Hope this helps. Ask if you have any other questions.
Mike
If you'd like the full story, here is the original NBD (new build day?) post: 
