There is no problem with Maxim's schematic. Of course, that schematic and your circuit are completely different, so there is little reason to think that the problem lies with Maxim's own application circuit. If you remove the everything involving the TPS2375, even then would Maxim's schematic not be suspect. Your layout would be suspect, however. Problems with switching circuits are, more often than not, a problem with the actual layout, and thus including only the schematic but not the circuit board layout is not useful and you will not get a solution to your problem. If the layout was fine, and the power source fine, then I would suspect that an ESD killed that chip, and try another one. If it still didn't work, only then would I consider contacting Maxim.
The chip has been out for 3 years. Countless engineers have designed it into products. Do you really think that your command of electrical engineering is so perfect that surely the fault must lie with Maxim's own datasheet? That this mistake has some how gone unnoticed by everyone to have ever used this chip in the last 3 years, and their success in using it was pure luck? Let's be reasonable. Of course there is no problem with the schematic in the datasheet.
I make a point of saying this because the single biggest downfall of many an engineer in my personal experiences has been a lack of humility. The moment one decides that a problem was not caused by yourself is the moment you become blind to the most likely source of human error (yourself). Sometimes you'll get lucky and the mistake really isn't yours, but all it takes is one time where it IS, and you'll never find it because you've decided you didn't make a mistake. Remember, we tend to be the worst person at spotting our own mistakes, so even if you feel 100% sure you haven't made a mistake, have another engineer look over things, or even better, have 2 or 3 engineers look. You'd be amazed how often anyone else, simply by not being you, is able to spot errors that you consistently miss. Making mistakes does not reflect poorly on your skill as an engineer - we all make them, all the time.
Now, normally any question involving a DC/DC converter would need to supply the layout, not just the schematic. But in this case, the problem is much simpler (and more easily solved, yay!).
Why aren't you using the provided power good output of the TPS2375 to turn on the MAX5033? That's the entire point of the buck converter's ON/OFF pin. UVLO is only needed when such a signal is unavailable. And this is likely the source of your trouble.
The problem is that the MAX5033, by using it's UVLO to decide when to turn on, instead of allowing the TPS2375 to control when the MAX5033 turns on. And the TPS2375 needs to have that control.
To comply with the 802.23af standard, there are certain requirements for inrush current. These requirements all revolve around the fact that there could be who knows how many feet of crappy, high-gauge ethernet cable between this power supply and the voltage source. With as little was 44V being available on the distant end of the cable, and significant resistance from the cable itself in between, current must be carefully controlled so as not to drop the voltage too far. This, by and large, is the whole point of using a chip like the TPS2375.
Buck converters are very input-centered devices. The primary ripple is at the input, not the output. Their input is trying to draw the maximum switch current when on, regardless of the output load. Loading the output varies the duty cycle of the switch being on, but it does not impact the magnitude of the current it is trying to draw while on, even if it is on only briefly. The average current will be much lower, but average current is not how the TPS2375 decides to do its job. Likewise, the 802.23af spec isn't talking about average current, it is talking about any current draw not exceeding various limits.
So, the MAX5033 is trying to turn on when the input cap is charged up to a mere ~6.6V (given your UVLO resistor values). This is while the TPS2375 is still in in-rush limiting mode. The power good output is also low - because the power is not yet good. No downstream circuitry should be drawing power at this point, yet the MAX5033 is definitely trying to. I would expect it to overload the current limit circuitry and prevent proper startup at the least. But, from the TPS2375 datasheet, page 13, under the POWER GOOD heading (which you, in theory, have read):
The TPS2375 includes a power-good circuit that can be used to signal
the PD circuitry that the load capacitor is fully charged. This pin is
intended for use as an enable signal for downstream circuitry. If the
converter tries to start up while inrush is active, and draws a
current equal to the inrush limit, a latchup condition occurs in which
the PD never successfully starts. Using the PG pin is the safest way
to assure that there are no undesired interactions between the inrush
limit, the converter startup characteristic, and the size of the bulk
Omit the resistor from the MAX5033's ON/OFF pin to GND, it is not needed (and 1 less BOM line is definitely a bonus), leaving the 1MΩ resistor as a pull-up (needed for the open-collector/drain output of the TPS2375). Connect the ON/OFF pin directly to the PG pin. Now the MAX5033 can be controlled by the TPS2375, and your circuit should work.
If it doesn't, then it's time to post that board layout!