Not all USB power supplies are rated for the current the rpi requires (i forget the number off hand). Anecdotally I had nothing but trouble using a name brand charger supposedly rated for the correct current, which disappeared the instant I switched to a $15 power supply specifically sold for rpi.

For the minor cost and potential headaches of brownouts and unexpected shutdowns, and given that you are asking the question, I would absolutely go with one made for an rpi.

I'd say the biggest issue is whether it's tranformer-based or a switching power supply.

Transformers are less efficient, and waste quite a bit more energy as heat.

You rarely see them these days, but I could see them being sold as the cheap version.

I don't know of a technical definition of either term that makes any difference. I think those terms are just used to describe how somethings used.

In my mind, power supplies are designed for one specific voltage and wattage output (maybe some variability in current, but only up to the max). Though not the same as things labeled Power Deliver or Quick Charge, which switch voltage levels between discrete, predefined voltages and current levels.

Power Delivery/Quick Charge type chargers are designed with circuitry that communicates with the device to negotiate voltage. Dumb devices (any device that doesn't request higher voltage) will just get the standard USB 5v (and whatever current the charger supports at 5v).

Power supplies are better. The pi requires 5V minimum, with a decent amperage. Most chargers put out 5V exactly. Unfortunately, a lot of usb cables are light on copper. This causes a voltage drop. Your 5.00V charger might only deliver 4.5V to the pi. This causes under voltage issues. It can also cause brownouts.

The pi chargers are designed to counter this. The cables have enough copper to not drop significant voltage. They also output 5.5V (top of the usb spec range), so can handle some drop. This makes them more stable in the long run.

I don't personally know of any difference except as marketing material. I have used devices marked under both names, for both purposes. The only exception I can think might apply here is if you get a charger that has a surge rating -- where it puts out a significant amount of power to force a rapid-charge. Anything suitable for powering a device will have a constant rate output, there is no internal circuitry involved like what is needed for monitoring rechargeable batteries (but that would likely be marked as a recharger or specifically as a battery charger, unlike a more generic phone charger).

Essentially what it comes down to is looking at how much amperage your device requires, and finding a charger/power supply that matches or exceeds this requirement. So like an older Pi worked with less than 2 amps, thus I have several 5V 2A micro-usb power packs laying around for them. It would work equally well with a 3A power supply or charger because the device itself still only consumes 2A of power. If I tried to use an older phone charger which only provided 1A of power, then the Pi would continually reboot because it wasn't receiving a suitable amount of power. Make sense?

A couple conversions to be aware of... 1A (amp) = 1000mA (milliamps), so that's an easy one. If you find a power pack that only tells you how many Watts it provides, divide that by the output voltage. So a 5W power supply that provides 5 volts will only give you 1 amp of power (5W / 5V = 1A).

Also note that a lot of Chinese sellers will provide an absurd rating for a given price, something like a 5A power pack for $8. What most people find when they open up these units is that the circuitry is barely even functional and more than likely to blow up the equipment you are trying to power. I tried one that measured fine on my multimeter, but as soon as I plugged something in the voltage dropped to almost nothing. These can easily be identified by being extremely light weight, for example something in a box the size of a laptop power supply but it feels like it is barely heavier than the plastic. Any reasonable power supply will have an obvious weight to it from the wire windings in the transformer.