Looks like its proprietry WSN http://en.wikipedia.org/wiki/ANT%2B WSN = wireless sensor network.
MiRF? Do you mean the Microchip MRF24J40 transceivers?
The Nordic Semi transceivers (nRF24L01 etc) are plain transceivers with some support for packet construction, and automated resend, but don't impose much protocol on you - higher level protocols can be stacked on top of the support they give.
The ZigBee is a whole protocol stack that supports a lot more functionality out of the box. I think the MRF24J40 transceivers support quite a bit of protocol in a mix of hardware and software (certainly the chip has a bewildering array of registers).
If you want a WSN out of the box, ZigBee is the obvious choice. If you just want raw comms go for a simple transceiver. If you want to play with WSN have a look around the web (I'd recommend jeelabs.org as an example of how to start building up a protocol suite) They use the RFM22B (that's not on the 2.4G band, note)