A relationship between two people A and B is the set of connections -- familial, social, commercial, etc. -- between them. The relationship of person A to person B means those connections from the one-way point of view of person A. In filling out a form, this is almost always a question about familial connection:
Q: What is (person B's) relationship to the applicant (person A)?
A: Person B is person A's father.
In other words person B's relationship to person A is that of male parent. Usually, this may be turned around from the point of view of person A:
Person A is person B's son.
That is person A's relationship to person B is that of male offspring.
Relationship with usually asks for the broader mutual connections between the two parties:
Q: What is (person B's) relationship with the applicant (person A)?
A: Person B and person A have been business partners for three years. They jointly own three commercial buildings, and Person A has sponsored person B for club membership.
Whether this distinction obtains for this particular application cannot be discerned without having the application posted.