If both devices are 100% accurate, the difference can be explained by the different load presented on both the coax and the antenna, more likely the coax.
You haven't told us the type of coax or the length or the transmitting power. What you are seeing is likely cable loss - some power not getting to the antenna (being absorbed on the way up) and some power not reaching the SWR meter on reflection (some power being absorbed on the way back down). The skew of the readings agrees with this. (WOW, I got to use "skew" in a sentence!)
Moving the SWR meter to the other end of the coax may or may not confirm this. Also possibly changing the transmit power.
A way to help most people understand this is a dummy load on the end of the coax. The SWR of a dummy load (in ideal conditions) is exactly 1:1 because all of the energy is consumed at the dummy load end; NONE is reflected back.
Some time back someone posted that they measured 1:1 SWR with a dummy load. As a joke, I posted "I hope so." Only a few people got it.