The tribological properties of the molecularly thin films of asymmetric ether (1,3-dimethylbutyl octyl ether, AE) and fluorinated asymmetric ether (1H,1H,2H,2H-perfluorooetyl-1,3-dimethylbutyl ether, FAE) were investigated. Friction forces and dynamic thicknesses (thicknesses during sliding) were simultaneously measured using the surface forces apparatus, and the effects of molecular asymmetry and fluoroalkyl. chains on the friction properties are analyzed. The friction forces (both kinetic and static) and dynamic thicknesses are larger for the AE film than for the FAE film. The two ethers exhibit stick-slip friction at low sliding velocity, but the stick-slip patterns are different. For the AE film, one stick-slip cycle consists of two or more spikes; a large spike is followed by one or more small spike(s) in the cycle. On the otherhand, regular stick-slip spikes are observed for the FAE film. The results suggest that the responsible friction mechanisms are completely different between the two ether films. The asymmetric shape of the AE molecule results in a variety of shear-ordered liquid structures in confinement, and the friction (stickslip) behavior follows the "phase-transition model". In contrast, the FAE molecule is rigid, and the shape of the molecule is rather close to a symmetric cylinder, which leads to a well-ordered two-layer film in confinement. The each molecular layer is strongly adsorbed on adjacent mica substrate and behaves as a fluorinated coating. The friction is governed by the molecular scale "bumpiness" of the fluoroalkyl chains lying on mica surfaces and basically follows the "cobblestone model". The advantage of the thin FAE film as a practical lubricant is also discussed.