Survey Question

About six months ago, we had a fence put up.  The fence company sent out a
licensed surveyor who found the mounuments,  The fence company then put a
string between the mounments to get the property line and then put the fence
3" on our side of the property line.  Our neighbor then did a survey and
their surveyor seemed to do something different and found part of our our
fence is like .5" on their property.  Their surveyor placed about 10 sticks
on his opinion of where the property line.  Six months later most of those
sticks are still there.  Our fence guy (big company with a great reputation)
did come back, check the survey and told me there is no way his survey is
mistaken.

Unfortunately are neighbors have been very difficult from the time we moved
in, and having a productice conversation with them would be a challenge.
So, before I try to talk to them, I would like to understand:

1.  The difference between the two types of survey's?
2.  Which type of survey is more accurate?

Thanks in advance.

David