The real difference is in the power handling. The actual power (and therefore current) provided by the transformer is determined by the load on the transformer.
The 1000 mA transformer can handle a load that consumes more power and also more current.
For the sake of simplicity, I'm going to assume a resistive load with no reactance, and therefore a power factor of 1. You will probably likely want to look up those terms if you're interested in learning more about transformers and AC power in general*.
If you have a 5W load (device you want to power) connected, it will draw 5W / 12 V = 417 mA (approximately). If you connected this load to either tranformer, it would draw 417 mA.
If you have, say a 10 W load, it would draw (or attempt to draw) 10 W / 12 V = 833 mA (approximately) - it would draw this if connected to the 1000 mA transformer. If you tried to connect this load to the 600 mA transformer then one of the following would occur:
The load would not function (because the voltage droops or sags too low to power the device)
The load would draw more current than the transformer can handle and possibly damage the transformer (depending on the transformer and load, could be immediate, or could slowly heat up and deteriorate over time)
The transformer fuses would "Blow" if installed and properly size for the transformer (to prevent the damage in #2)
This is a simplified explanation. Most transformers can actually handle a little more than what their rating states - so they won't always get damaged, but you should always stay within the manufacturer's specifications to ensure maximum life of the equipment.
*If the loads are reactive and have a pf other than 1, you cannot simply divide power by voltage to get current.