In practice, what is the difference between 2/2 and 2/4? Doesn't the emphasis fall in the same place either way?
Practically speaking, there's no difference between writing a piece in 2/2, or writing it in 2/4 and halving all of the note durations. They would sound exactly the same.
The difference is largely convention and tradition. Marches and upbeat musical theater numbers are traditionally in cut time, but most other styles of music prefer the quarter note to take the beat.
In theory, the exact same music could be written in either time signature, either with the notes being half the (written) duration in 2/4 and then played at half the speed (4 eighth notes in 2/4 taking the same amount of time as 4 quarter notes in 4/4 time) OR with identical note durations and twice as many measures of 2/4.
In practice, time signatures often indicate some level of implied accent or emphasis. For example, the ^..^..^. (12312312) rhythm common in rock music might be more accessible to the musician written in 8/8 time than 4/4 time, and is certainly more difficult when broken across 2 measures. Specific to 2/4 time, we may expect an accent on every other beat (the down beats) which is common in marches. 4/4 time may have the expectation of an accent on the downbeat, but may have secondary accents on beats 2,3 or 4. In any signature, accents may or may not appear anywhere, but composers want their music to be accessible to musicians and so they make choices to facilitate the playing of their piece.
On a practical level, they are exactly the same in performance. However, 2/2 time is somewhat of a leftover from early music (chant, etc.) that used the open noteheads of what we now call whole and half notes (since they're almost always referenced now to at least 4/4 time where they'd have the temporal value of a whole or half measure).
I personally believe that note values and time signatures are still important to the performer, as they do implicitly convey a feel. Very good examples can be found in much of Stravinsky's music, where he used note values and signatures for this exact purpose - as is found in Le Sacre du Printemps, for example. There are also 19th and 20th century examples of liturgically based works (i.e. Requiem masses etc.) where the composer still chose to use half notes as the beat. I believe Verdi, Bernstein and Stravinsky all did this to some extent.
2/4 and 2/2 are principally about the same. As a performer however, I tend to have different feelings about them.
Starting from the "standard" 4/4 with its alternating strong/weak accents, 2/4 has not-really-alternating strong/strong accents on the half notes while 2/2 feels more like leaving off an accent on the second half note as compared to 4/4.
So as a performer I lean towards giving 2/4 more of a "pulse" rather than 2/2 which is a bit more "measured".
One way to look at the difference between 2/2 and 2/4 is to start from the assumption that someone meant something by the choice, and back out what they could have meant. After all, if it really didn't matter, we wouldn't waste time with a notation that discerns the difference.
I would start by looking at the difference between 2/4 and 4/4. At the most shallow level, each one is a bag of notes, to be played in order. Little difference should be apparent. However, we are well aware that the stresses in 4/4 are highly diverse. Most of the time you have a strong stress on 1 and a lesser stress on 3, but there are dozens of other stresses which are associated with 4/4.
4/4 is kind of the default these days (feel free to disagree for historical pieces). So that means the choice of 2/4 had to mean something. The most logical meaning is that, instead of a strong stress on 1 and a lesser stress on 3 (or any one of the other patterns), the composer is indicating a strong stress on every other beat. This, of course, is not an answer to your question, but it does lead in the direction.
Now 2/2 is a funny time. It's not 2/4 and it's not 4/4. If a composer is using it, it's either very historical (in which case the meaning will likely differ between composers), or its something related to the numbers themselves.
Consider that, while 2/4 clearly wants a stress on every other beat, it points to the beat being on the "quarter note." This does suggest a sense of finality that should occur every other measure, as we sum up to "one whole note." It's not 4/4, so we don't want to see that finality in the stresses, but we do expect to see some level of finality appearing in even numbered measures.
2/2 would erase this assumption. While 4/4 would abhor a phrase ending in 14 beats, and 2/4 would treat it as the "spice" intended by the author's time signature, 2/2 just treats it as "the way it's done." No need to draw attentional attention to what we could consider an "unusual" phrase to the modern ear.
How much does this interpretation of the meaning of 2/2 matter? Depends on the piece. If you find lots of strange timings, it may explain what the author intended. If you find it actually fits a very traditional timing pattern, it may be as simple as "2/2 was the historical way the song was notated." However, hopefully this answer suggests directions you can look in when you look at your piece of music and decide how you want to interpret the notes, all written in their universal crisp sterile black toner in uniform shape on bright white backgrounds.