Hi,
For example, one op amp model might let us only change the open loop gain, while another will allow us to change the gain bandwidth product too. This second option requires additional passive components while the former only requires a voltage controlled voltage source.
There are a lot more options here though. Another model might let us change the input offset on either input terminal, using two constant voltage sources or may be even more complex to follow some function of the power supply voltage.
It is interesting that a voltage controlled voltage source does not require a power supply of any kind, neither plus nor minus, and it can still generate a plus or minus voltage. This is like magic though and must be dealt with when doing a power study where the efficiency might come into question with a real op amp when with a simulated one it might not. If this is important, then a more sophisticated model should be used.
A really ideal op amp would really have infinite gain and infinite bandwidth, so maybe it is just better to call these other ones "behavioral" and each added option mimics an additional behavior.