Electronic – Why does capacitor polarity matter

"Film Capacitor" typically denotes polyester or polymer film as the dielectric - as another answer points out, metallized film capacitors are the same thing: A metallic coating being applied to an extremely thin polymer film, to create the conducting electrodes of the capacitor.

In general, ceramic capacitors are somewhat non-linear in their frequency and voltage responses, compared to film capacitors. Another issue with ceramic capacitors is that they tend to behave as microphones, thus picking up ambient sound and modulating the voltage across them accordingly.

Also, for smaller values (a few pF), ceramics were more commonly used, while larger values were somewhat likely to have film considered as an option - or at least that was how it used to be, before capacitors became so inexpensive with the advent of SMT, that the price difference became negligible except for huge volumes.

Both film caps and ceramic ones are non-polarized, so that isn't a difference.

The two capacitors used in the LM317 typical application are described on page five of this datasheet. An identical schematic to the one you provided is given:

Along with a helpful note about why the capacitors are needed:

Note: 3. \$C_i\$ is required when the regulator is located an appreciable distance from power supply filter. \$C_O\$ is not needed for stability; however, it does improve transient response. Since \$I_{ADJ}\$ is controlled to less than 100 μA, the error associated with this term is negligible in most applications.

The best practice, for either capacitor, is to always include them. Specifically for \$C_i\$, place it as close to the input pin on the LM317 as reasonable.

If you have determined that you know what you're doing, you can decide to omit or alter the suggested schematic. In determining that, for \$C_i\$, an appreciable distance in on the order of centimeters, but you should also factor in the level of noise you expect on the input supply and how much the load the LM317 will be changing. For \$C_O\$, you can likely omit it if you have decoupling capacitors at the inputs of any ICs and relatively high current draw devices, anywhere things are going to be changing how much current they're drawing in a short amount of time.

Do check out the related questions and answers regarding how to use decoupling capacitors and how to select the type of capacitors to use.