They give different results indeed. Let us know how you've got this issue solved in the end.
I've noticed an error in you data however (which, in fact, makes the difference even greater):
Saturn PCB allows you to set base copper thickness and plating thickness independently. So, according to your picture, when you set base copper weight as 35um AND plating thickness 35um then you essentially calculate impedance for 2oz copper (not for 1oz as you do in Altium). The app shows you this at the bottom: "Total Copper Thickness: 70um".
So in Saturn app it should be set either as '35um base copper + bare PCB' or '18um + 18um' - both variants will mean 1oz copper (finished thickness), however, since internal layers are not plated, they would also mean 1oz internal plane or 0.5oz internal plane correspondingly. See the 'General Settings' section here
https://www.saturnpcb.com/toolkit_help.htm for explanation of those options.
Cheers!