Saving to the disk and Write Changes to Device are two separate things - you have to do both if you want to save a copy of your settings to computer as well as transfer settings to the LT.
Using the Rack Gizmo's Write button vs the LT's IA Store button is a different story. Which one you should use depends on the answer to this question: do you have IA buttons on the LT that control each individual Rack Gizmo loops and switch (at least the ones you're using)? If so, you should never use the Rack Gizmo's Write button, it won't have any effect. Use the IA buttons on the LT and IA Store to select Rack Gizmo loops and switches instead.