How to determine hardware/software compatibility? Currently, I'm using Greiman's SdFat Library with Adafruit's Data logging Shield. 

I'm thinking about switching to the Adafruit micro SD Card Breakout. I imagine that it will work as both products are made by Adafruit, and the board schematics look similar. However, I don't know what to look at to be sure they are the same. 

Is it enough that they both use the HC4050M chip, or is there something else that I need to be aware of?

Short of buying one and testing it. What do I need to look at to verify that these two boards will work with the library?
The HC4050M is just used for logic level translation (SD cards are 3.3V devices). It is "enough" that you are talking to an SD card using an SD card library. The SD card is really all the hardware there is - everything else is just interfacing fluff - completely transparent to the software.