I wrote the Forgotten Heroes monk, so my comparison to Ari's martial artist has the benefit of being informed at the cost of being biased!
That said, I think both books are must-haves for 4E players who want to include any of these classic archetypes in their game. The approaches taken by the APH and Forgotten Heroes are different enough from one another (and from the way the PHBII is likely to go) that you could easily have a group with an APH martial artist, a FH monk, and a PHBII monk without any of them feeling like the others had stolen their thunder. Or, you could take your favorite elements from all three versions and recombine them into your own vision of the ultimate monk. It's awesome that iconic character options like "master of many weapons, reliant on none" or "lightly-armored, heavily-muscled fighter" have gone from having no good options to getting multiple books worth of love!
I think an easy comparison between the two classes you mention, Jack99, is to say that the FH monk is a striker in the mold of the PHB rogue, while the APH martial artist is a striker along the lines of the PHBII barbarian (or at least its preview).
Playing a rogue is all about tactical choices. Working to set up combat advantage rewards you with bonus damage, but requires you to take risks (like moving into the thick of things to get flanking) or give up other options (like spending a round hiding so that you can pop out next turn). The rogue is my favorite striker because its tactical choices are deeper and more meaningful than the warlock's or the ranger's, so I designed the FH monk to offer a similar depth of choice during play. The monk's bonus damage depends on tactical choice; like a rogue, you have to work to set up the conditions in which you get it, and each build of monk has a slightly different set of special conditions reflecting their fighting style. Stances are another example - you have four stances available, and have to choose which one to use each turn. The drawback is clunkiness, or a higher learning curve; the advantage is increased options on your turn, and the fact that each stance option can be more powerful because they're either-or instead of always-on.
Playing a barbarian looks like it's about ease of play rather than tactical choice. That's not at all negative; some people just want to hit things real hard without worrying about fiddly rules like combat advantage, and those people weren't well-served by the PHB melee classes. I see the APH martial artist as offering a similar straightforwardness. The barbarian doesn't need to set up any tactical situation to get striker bonus damage; he's got extra dice written right into his powers. The martial artist is similar; as long as he can hit you, he can get bonus damage from a quick-strike. Likewise, the martial artist doesn't need to choose between fast movement or good armor class; they're both always-on (and appropriately balanced so that neither is as powerful as the FH monk's either-or options).
If I was going to point to one advantage of the FH monk, it's in the way it handles weapons. As I was watching Hong Kong action films for inspiration (both the '70s ones that would have inspired Arneson's monk in the Blackmoor supplement and more modern examples), it seemed to me that the hallmark of this fighting style wasn't that it was weaponless - instead, the hero fights skillfully with whatever weapons come to hand until they break or are hurled away, and then continues fighting bare-handed at no disadvantage! To reinforce this genre feel, I put in a number of powers that involve the monk throwing or shattering a weapon; added secret techniques so that they could switch weapons or fight unarmed without losing the benefits other classes get from magic weapons; and worked to ensure that a monk player wasn't at a disadvantage if they followed the spirit of the class and didn't invest in the kind of weapons monks aren't "supposed" to use. I'm not sure the APH martial artist hit this last design goal; at first glance it seems to me that up until 21st level, your martial artist would do well to take the Weapon Proficiency (bastard sword) feat and forget about all this unarmed strike nonsense.
If I was going to point to one advantage of the APH martial artist, it's the way it points to powers from the PH that suit the class. I've seen some reviews that sniff at this, assuming it's just word-count limits, but I think it's actually a great design decision. First of all, I think the PH precedent of not sharing powers between classes is lame; that clerics can't have dispel magic because it's on the wizard list makes no sense to me. Second, I think Ari's example encourages the way I personally prefer to approach 4E class powers, which is as a set of pre-made parts I can kitbash into something that suits my personal tastes. So I think it's awesome that the martial artist can tumble like the rogue, and certainly my personal version of the monk is going to be able to choose the serpent bends aside power from the APH!
