Please correct me if I am wrong: It looks to me that the top image has first-order elements and the second image has second-orders element. If that is the case, it could cause some difference, providing the mesh is not fine enough.
To answer your second question, matching simulation results with the experiment has been a very broad topic and it could be infected by so many variables. Here is what I would check:
1. From representing the physical test point of view, does the model have accurate enough boundary conditions? Ex, simply supported? constrian rotation? Line loading? Point loading?
2. Does the material model represent the physical material well? Does Young's modulus match the experiment? Does the experiment show more complex behaviors like anisotropic behavior or material nonlinearity? If you are modeling a rebar-reinforced beam, does the model considers the interaction between the rebar and the concrete well?
3. From the numerical modeling point of view, Is the mesh fine enough to provide enough accuracy? Is Large deflection turned on?
These are just my thoughts from my experience, but hopefully, it can give you some ideas.