You know, SWR is different at different places along your coax so when you see different measurements in different components, that is normal. This is more obvious as the SWR at the antenna is higher and almost negliable if the antenna is flat. You can experiment by pulling out the coax that goes into your tuner (I assume goes directly to your antenna) and plugging that into your K3 as an interesting test. While I wouldn't expect them to be exact, they should be much closer and prove to you that SWR is different at different places up and down your coax. You could also add a random length of coax to your antenna coax and I'll bet your analyzer gives you two different readings (compairing with and without extra coax). If you want REALLY good SWR, add 1000" of RG-58 and try 10M...just kidding, because at that point, there won't be much signal left even though your SWR will look great!
I'll leave it to one of the engineers on here to explain why but as an appliance operator, that is what I've observed in practice. ;-)