Each software may be using a different reference state.
To check if this is the case or not you can use a simple trick.
Just try and calculate the enthalpy or entropy difference between two states from the two software, (the same two states). If everything other than the reference state is the same, this should give you the same answer.
For the conversion method you mention, you can calculate the difference of entropies and entropies as Mohamed Abubakr Hassan says and then subtract or add it where you need it.
You can also find out the EES reference state and enter it in ASPEN to get the same values.