What is the difference between a radio antenna and a capacitor?

All elements of a circuit will radiate when there is alternating current involved. Typically though, at the frequencies involved, not very much. It is something that circuit designers must be careful of though.

Capacitors themselves typically are encased in some sort of material which will damp the stray radiation, especially if it is a conductive material and hence makes a little Faraday cage.

First of all, the photon picture isn't the best way to visualize and model the operation of a radio-frequency antenna. RF antenna engineering is best performed and understood (IMHO) by means of a model in which high frequency AC power flow through a wire creates electromagnetic waves that propagate away into space.

For this to occur, the antenna impedance must be well-matched to the impedance of free space. As such, every antenna possesses a certain (small) amount of capacitance and inductance- so one plate of a capacitor is not an accurate model of an antenna (although capacitor "plates" affixed to specific points of an antenna are commonly used to add enough capacitance to it so that in conjunction with the antenna's intrinsic inductance, the antenna is resonant at the frequency of the transmitter connected to it).

Now note that in a capacitor with two plates, the electric field gets concentrated into the gap between them- by design, you don't want to "waste" the field by letting it leak out of the gap. Some leakage is inevitable, and at very high frequencies, even a well-designed capacitor will leak enough to radiate small amounts of electromagnetic waves.

I think your question might be valid in some non-technical sense. The truth is, the signal does lose energy. As someone previously stated, it is the power of the signal and the impedance of the atmosphere that matter. As the broadcaster of the signal, you have to push enough power through the air to send the signal. If you want to think of these bodies as the plates of a capacitor, you can. But that’s not really what people mean when they talk about a capacitor. But you bring up a valid point. Capacitance (and inductance) happens all the time and in all directions.