Advantages

Much better thermal stability, as the mirror spacers can be fabricated from dimensionally stable materials and the optical path through air is relatively temperature-insensitive
Mechanically more robust if the FSR is large (small mirror spacing)
Small FSR values are easier to achieve (large mirror spacing)
Both mirrors can be coated simultaneously, resulting in near-perfect matching. This is particularly important for broad waveband or large mirrors, where mismatching of reflectors can lead to serious etalon transmission losses
Potentially better effective finesse, as the issues of mirror flatness and parallelism are addressed separately
They can be pressure-tuned

Their complexity of construction tends to make them more costly to produce than comparable solid etalons
Air-spaced etalons are bulkier than their solid etalon equivalents
Because of the optical contacts air-spaced etalons are in general less rugged than solid etalons
Hard coatings can cause a net distortion of the reflecting surfaces, resulting in a degraded effective finesse
Losses in the anti-reflection coatings may result in lower transmission than for an equivalent solid etalon

As a result of their simpler construction they are cheaper to produce than equivalent air-spaced etalons
Solid etalons are more compact than their air-spaced equivalents
Hard coatings, which can distort flat substrate surfaces, have no net effect when the same coating is applied to both sides of a solid etalon
They can be temperature tuned
Solid etalons are more rugged, unless the mirror separation is very low (very high FSR)
Potentially they have better transmission characteristics (no anti-reflection coatings)

Solid etalons have poorer thermal stability. This is partly due to the fact that the linear thermal expansion coefficient of fused silica is several orders of magnitude higher than those for the low-expansion materials used to fabricate spacer blocks for air-spaced etalons. But a more important factor is that the thermal coefficient of refractive index is about twenty-times higher than the expansion coefficient
There is a practical limit to the thickness of a freestanding solid etalon. For small apertures this is about 50 microns. In deposited solid etalons there is a maximum spacer thickness limitation of about 5 microns. Generally speaking, this means that mirror spacers of intermediate thickness are inaccessible to solid etalon designs (although equivalent air-spaced etalons are possible)
Potentially, solid etalons have poorer transmission characteristics due to mismatching of the reflector coatings
There is more scope for defects in solid etalons, as high flatness and high parallelism have to be achieved simultaneously in the surface polishing process