What is difference between 2 transformers rated to give output the same voltage but different current (ampere)?
Short answer is NO.

The difference is that 1A transformer can provide more current if necessary.

On the same load they will provide the same current if current needed is below 600mA. If the load is of lower impedance (resistance) than higher rated transformer would function without issue, but 600mA version would have voltage drop and it would heat a lot. Also correct working above declared specification is not guarantied, so if you load 600mA transformer with 1A (for example) it could fail.

On simple load, needed current can be calculated using Ohm low: $$ Current = Voltage / Resistance $$
You can read more here https://en.wikipedia.org/wiki/Ohm%27s_law