The propagation speed is often expressed as the velocity factor of a medium - the fraction of the speed of light in vacuum that you get.
On the physical side, light going through a medium is slowed by the medium depending on its refractive index. Fiber has the added 'problem' that the core requires a slightly higher refractive index (optical density) than the cladding to properly guide the wave. The effective propagation speed is the speed of light divided by the refractive index, or the velocity factor is the reciprocal of the refractive index. Most fibers have a velocity factor of or close to .67.
Copper is a bit more complicated. The actual electrons are not moving substantially, a high-frequency signal is rather an electric wave (field fluctuation) flowing through the cable - somewhat comparable to sound in air. The propagation speed of this wave surprisingly does not depend on the conductor alone but on the combination of the conductor and especially the insulator (its permittivity) because the wave needs to propagate through the latter as well. The effective propagation speed is the speed of light divided by the square root of the permittivity.
For copper, a velocity factor of close to 1.00 is possible by using air as insulation as with special coax cables or open ladder cables. Copper network cables range from .77 (RG-8 for ancient 10BASE5) to .585 (Cat-3 for 10BASE-T) with the common Cat-5e and Cat-6 at .65 (=slower than fiber).
As has been pointed out, in practice, there are lot of other factors contributing to the effective propagation delay such as transceiver technology, encoding overhead, forward error correction and possibly retransmissions. Especially the encoding overhead can vary significantly from any-length fiber (<.1 μs delay) to voice-grade twisted-pair copper using interleaved ADSL (~20 ms = 20,000 μs delay).
So, the propagation speed or the velocity factor isn't usually critical.
As to fiber "being better" - it's got higher performance for sure, but "better" depends on your requirements, including cost.