I have used both, but have more experience with Permalyn. LMF Permalyn sealer, by all accounts, is an excellent deep penetrating sealer and can be used without the somewhat thicker Permalyn finish. Permalyn finish is fairly glossy, and if you have left any minor sins on the surface of your work, it will be quick to point them out. On my last rifle, I started with a single heavy application of LMF sealer, and followed with several applications of Chambers oil over this with no further Permalyn finish. This created a very nice soft satin finish which I personally find much more pleasing than the hard gloss of the Permalyn finish. (This was the first time I used Chambers oil, and I was very happy with it.)
Chemically, the two finishes are not the same. Their actual chemical differences have been pointed out on previous posts here.
Gregg