I've DM'ed both, extensively, over the last few months. At least 40 hours play with each.
In my experience, the "broken-ness" of the moon druid is overrated. I've seen two die already. Here are my observations:
1) The brown bear is the shape-of-choice for the moon druid at low levels, sometimes interspersed with the tiger.
2) The huge buffer of hp offered by the bear is very nice, and they have formidable damage output. But at 4th-5th level, this was not in any way outpacing the party fighter or monk. The monk was dropping 4 attacks per round with ki flurry, for 1d6+4 per hit and a better attack bonus than the brown bear. The dual-weapon fighter was not being over-shadowed either. Their Extra Attack action at 5th level kept them on par with the bear, and their AC was far superior. Action Surge was pushing them in front. As for the spellcasters, none of them felt they were being overshadowed. The bear can't fireball. And that, to me, is the most important consideration by far: do the other players feel like they're living in the shadow of the druid? For my group, the answer was a clear NO. As DM, if I feel it's unbalanced, I can always sic more monsters on the bear, have it soak more attacks, etc. But it's the other players' opinions that matter most. And they thought it was just fine.
3) The bear's big vulnerability, of course, is AC 11. In many respects, that's a real risk for the druid. Sure, they have a big pool of temps, but excess damage rolls over the top. I had a player druid sweating for nearly 2 minutes at the table, trying to work out if it was too *dangerous* to wildshape (or stay wildshaped) when facing down a group of hobgoblins. Hobgobs, if you don't know, have poor attack rolls (+3) but do hefty damage with martial advantage (2d6+1d8+1). The same is true of many other creatures (most small creatures with poison, for example). Sometimes, it's better to have the high AC and only be hit once, rather than get hit multiple times and have a bunch of the damage roll over onto your regular hp. The moon druid has seen too many times when his bear form was on 12hp and got tagged by low attack rolls for 30+ damage. He'd rather have been in barkskinned regular form and been missed by everything.
4) The real kicker, though, is the lack of spellcasting in wildshape (at least until higher levels). If the bear can act as a Defender (i.e. soak most enemy attacks), then no problem. But if it's a general melee versus multiple opponents, the other PCs are going to be lamenting the loss of healing.
5) How did our 2 moon druids die? One ran into a medusa at 5th level. Surprise! You're a statue. No amount of bear hp is helping there. The other one simply got overwhelmed by an owlbear at 3rd level. He went into wildshape, and found out the hard way that it's not 4th edition any more. All the analyses on DPR and survivability-over-time tend to go out the window when a combat can be largely over in 3 rounds. The druid wildshaped and moved to "take the beats", protecting the rest of the party with his huge buffer of HP. But then he took a hit and a crit (the owlbear, of course, needs only a 4 to hit the bear's AC...). Blam - no more wildshape. Suddenly, he's a spellcaster on the front-line. Over the next two rounds, the owlbear proceeded to turn the druid into a glove puppet. The player's next character was a fighter with AC 19, second wind, and a lot more hp. He lived longer.
6) Moon Druid Anecdote: A few sessions before he got petrified, the 5th level moon druid player was adventuring in the desert. He came across an oasis with a large pool of water. The party wondered if there was any loot at the bottom of the pool, but nobody was keen on wading in. So, the moon druid turns into a shark and leaps into the pool. In the middle of the Sahara (effectively). At that point, a trio of giant scorpions erupted out of the sand nearby and proceeded to lay into the rest of the party. On his turn, the moon druid leaped out of the oasis pool in shark form (Athletics / Jump) and chomped down on one of the giant scorpions. We now have a shark threshing around on a sand dune in a struggle to the death, with an 8'ft long scorpion in its jaws. Awesomesauce. You know what that player's biggest disappointment was, about dying at 5th level? Because they weren't going to have the opportunity to turn into a dinosaur (allosaurus/plesiosaurus) at 6th level.
Druid Analysis: It's fine.
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Let's talk about the monk. We've had 3 of them so far. Probably the most popular class in our group. Everyone wants to be Bruce Lee / Jackie Chan. Or an airbender. No Way of the Shadow yet, strangely. Ninjas were so popular back in the '80's...
1) The monk is a tough one to analyse. Firstly, it's worth pointing out that they've all since died. The aforementioned medusa, a bandit ambush, and a lesser earth elemental. They're not invulnerable. Low-ish AC, mid-ish HP. Even more than the druid, this one lives or dies by the Short Rest.
2) It's worth noting that I'm a fairly tough DM. If the guidelines say that a typical adventuring day has 6-7 encounters, and expects a maximum of 2 short rests in that time, then I'll adjust the module to make sure that happens. The princess dies in only 3 hours! Wandering monsters (i.e. the next 1-2 keyed encounters) come looking for YOU when you try to rest! You can't rest in a sandstorm! The DM is NEVER subject to the desires of the characters when it comes to Short or Long Rests. You run the story, you set the challenge. That's your job, dammit.
3) So, the monk has to be a bit judicious when using his ki points. All three monks deaths in my campaign occurred when they were out of ki. Why? Because of ki=bonus_dodge. It's not about the 4 attacks per round, or the stunning strike. It's about dodge as a bonus action. Giving all attacking creatures disadvantage for a round is mad strong, especially when a fight might only last 3-4 rounds. Of course, as DM, you can be judicious about that as well. Unless the enemy has no other choice, it's pretty obvious that the monk is ducking-and-weaving. Go beat on someone else for a round, instead. I'll generally run it according to my opinion of threat. If the rest of the party is already under heavy pressure, then I'll have some of the foes attack the obviously-dodging monk. If it looks like they're handling the combat just fine, then I'll ignore the monk and beat on someone else for a while.
4) I found the airbender monk a bit weaker than the fists-of-fury monk, but I can't tell if that was mechanics, build or circumstance. Certainly, the fists-of-fury monk can do some decent damage output. And they have more maneuverability and flexibility than the party fighter. Again, the acid test: did the other players think the monk was too weak? No, because they keep building them. Do they think it's too strong? No, because they keep dying - and their edge over the fighter or the wizard isn't in damage; it's in "cool-ness" and secondary characteristics (speed, etc).
5) Monk Anecdote: Hoo boy. This one really rocked the table. I've laid down a multi-room battlemap, covering maybe 4ftx4ft of my gaming table (using 3d Dwarven Forge). Various rooms and corridors. The party has cleared most of these, and is fighting a bunch of yuan-ti in a corner room. Party wizard casts haste on the monk. And here we go... Monk has base 40' move at 5th level. Haste doubles speed, and gives extra action. Monk burns ki to take Dash as Bonus Action. This is what happens: Monk attacks yuan-ti twice, smacking it around (Attack action). He then moves 160' (32 squares, Bonus Action from ki), passing through every other room in the laid-out dungeon, running down corridors, tearing through doors... runs up behind the same yuan-ti and attacks it again (extra Haste Attack action). He did a complete circuit of the dungeon between punches. Awesomesauce.
Monk Analysis: It's fine.
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So, I have no issues with either class. Other than the fact that, in the hands of a competent DM and a creative player, they may cause dangerous levels of Awesome Overload at your table. Remember to wear goggles, for your own protection.