Switching between age and grade norms switches the makeup of the group you are comparing your examinee to. Standard scores provide you with an indication of where your examinee stands/falls in comparison to their peers.
Example: You are testing a 9 year old in the 4th grade.
When using age based norms for a 9 year old, you are comparing them to same-aged peers (other 9 year old's) in the norming group.
In turn, when you use grade based norms for a fourth grader, you are comparing them to same-grade peers (other fourth graders) in the norming group.
Whenever you change the norming/comparison group, there is the reasonable expectation that your standard scores will also change. This is because where an examinee falls when compared to same-age peers may be the same as where they fall compared to those in their same-grade.
If you were trying to compare your height to your coworkers, you might be in the middle (3 coworkers taller than you and 3 shorter than you). But if you went to the grocery store and tried to compare your height there, you might be a different ranking (2 other shoppers taller than you and 6 shorter than you).
Your ranking (which is what a Standard Score gives us) changes depending on who you're comparing the examinee to.