The stiffness update in a NL simulation consideres both geometric and material non linearity.
In linear static none of above is considered. If you turn on Large Displacement, the geometric stiffness will be updated. Material nonlinearity can only be applied in NL sim.
I think you did nothing wrong. As Janko Stellaard said , these are different solver. So these answers must be different.
ı dont know the dimensions of your parts. If relative displacents are so small, you can accept linear static.
Or not use nonlinear.
Thank you, Janko Stellaard and Seckin Uslu for the replies. It seems that it was the large displacement option that caused the differences between the simulations.