<p>Some electronics come as a single PCB. They have CPU and everything on just one board.</p>

<p>Other electronics are a shield for an Arduino. So these are always two boards. The Arduino and the other PCB.</p>

<p>You probably don't want to exchange the Arduino unless it is broken. Does it break that often or are the two boards just the predecessors of the one board solution?</p>

<p>What are the benefits/downsides to having one/two boards?</p>

Many 3d printer motherboards are based on Arduino/Atmega microcontroller and just add some stepper motor drivers, MOSFETs and such in a single board. That explains why you use the Arduino IDE to update or modify their firmware.
Now why you would want to use an Arduino + an Arduino shield board like RAMPS? Well if you're not good at electronics, are happy with the cost of your own board, don't care much about upgrades/modding beyond what is possible with your board, maybe you shouldn't, it might be overwhelming.
There are several advantages with using Arduino with a shield for your 3d printer, "two boards are not just the predecessors of the one board solution", no.
1) It is moddable/exandable/upgradable/has replacable parts.
If your printer came with its own motherboard that doesn't have additional or enough pins to add more fans, enclosure lights, a second extruder, an LCD and you want to, it sucks. RAMPS can do that, it has plenty of extra pins.
It is upgradable. You want to replace the stepper drivers with a new one? Or you accidentally damaged the one you have? Fear not, you can just replace that instead of the whole motherboard.
Think if it like other motherboards being PCs on which you can't change the CPU, RAM and GPU.
2) It is here for a long time, you will be able to acquire one for a long time.
There isn't just one company making RAMPS or similar sheilds. Sometimes 3d printing companies go out of business or stop producing your particular motherboard. RAMPS is likely here to stay. Because Arduino is very likely here to stay.
3) As said above, not just one company owns or makes RAMPS or other Arduino shields. Besides the possibility of your motherboard not being produced anymore, there's also the advantage of not being at the mercy of one companies pricing and shipping policies. This is true for some other boards as well though.
4) It is possibly cheaper than what have you. I don't want to post a link, but one company right now is selling their derivative of Printrboard for like $180. Check the cost of Arduino Mega, stepper drivers and a RAMPS board yourself, quite a difference. That said, there are some boards which are close in price.
5) It's an Arduino. Why is this a good thing by itself? Because many people who have a 3d printer are tinkerers/makers and they already use Arduino for other projects. It is open source with a rich library to control many things. The modding and upgrades for your 3d printer which can be done with Arduino is another level higher. Or it can be a good learning experience for your future Arduino projects. If on the other hand you already use Arduino and are experienced with it, you might use it just because you know how to control it/fix it better than some specialized board you haven't seen before. Plus for a guy like you your 3d printer's motherboard will be an "off-the-shelf" part, that's nice, right?
Video on RAMPS:
https://www.youtube.com/watch?v=FYJn6FuWOv4