I like the Note 3, the SGS4 underwhelmed me for it being the flagship, the G2 is a better looking and better performing device IMO and I have spent a lot of time with both. I find Sony's industrial design and UI to be very underwhelming really. Galaxy J isn't available outside Korea right now but the spec looks nice.
IMO:
1. G2 / Nexus 5
2. Note 3 / SGS 4
3. HTC One Max
4. Sony Z1
Compared to the difference between the S and S2, and the S2 and S3, the S4 isn't different enough to the S3 really. I find the design to be uninspired and I dislike their insistence on filling up the phone with useless bloatware and failed clones of Google Apps. Sure, other OEM's do that too but Samsung is taking it to the extreme. ChatOn, Samsung App store, all that shit can fuck right off. Nobody uses it or likes it. They ought to stick to hardware and leave the software to Google.
The Nexus 5 isn't as great as the G2 though hardware-wise. The Note 3 is basically a bigger G2 with even faster chipset, better camera (minus OIS), battery and screen - I think it's the best phone out there right now if the size doesn't bother you. Samsung's software is also superior to LG's software, but I agree about shit nobody needs being installed there. But I find it true even for the Nexus devices these days as well - I don't want half of preinstalled Google apps as well.
I think the S4 was a somewhat meaningful upgrade because it really does look better, the chipset is much better, the battery life is much better and just overall everything else is somewhat upgraded enough to make count imho - even the smaller size while having a bigger display was a huge upgrade alone.
It's not the newest phone at the moment, but it's still holding strong as one of the top choices.
I don't think the HTC One Max is an interesting phone though. There's nothing good about it really except of a case some people seem to like, minus the fingerprint scanner that spoils the design. I think the Z1 was a much better phone - the camera, waterproofness and chipset made it that way in my book.
I'd say right now the best phone depends on your needs. Hardware wise, it's the Note 3 followed by the LG G2 and Xperia Z1, then the Nexus 5/SGS4/HTC crowd - the Nexus 5 has a great chipset but there were other hardware cuts made to it..
However, bang for the buck ratio and stock Android needs get satisfied best by the Nexus 5.