I cannot figure out the actual difference between these two beats.
Isn't it sufficient to have only 2/4 (or 4/4 )?
Go for a walk. Count each step, in twos or fours. Tread heavier on the ones.
One two One two One two One two
One two three four One two three four
They feel different, don't they? This is the difference.
And yet there is an equivalence between them. Walk at the same tempo, but count to four twice as fast, so you're stepping on the One and the three.
One (two) three (four)
This feels like the 2/4 rhythm again, except for the in-between-steps counting you're doing.
This sort of equivalence comes up all the time. In environments where musicians play without sheet music, it's quite possible that the drummer thinks he's playing in 4/4 (with triplets), while the guitarist thinks he's playing in 12/8. In this example, what the drummer calls a triplet, is three 8th notes for the guitarist.
When a composer scores their music, they choose the time signature that they believe conveys the feeling most clearly to the musicians, while being easy to read.
- 2/4 with 4 quavers to the bar at a slow tempo.
4/4 with 4 crotchets to the bar at double the tempo.
-- that's mostly a readability choice.
The other answers are all essentially correct, but I think a critical point is missing.
There aren't just fully "strong" and fully "weak" beats; there are also beats of medium strength (and other varieties). 4/4 is most commonly emphasized like this:
ONE two three four
Note the half-accent on the third beat, different from what slim mentioned. If you're playing triplets, that works out to:
ONE two three four five six
Slim was right in saying that 4/4 with tripletted quarters indistinguishable from 12/8 with normal quarters, but it's only true because of that emphasis on the third beat. 12/8 is emphasized like so:
ONE two three four five six seven eight nine ten eleven twelve
With quarters (combining pairs of eighths) that works out to:
ONE two three four five six
Which is exactly the same as the above 4/4 with triplets.
Now to get to your specific question: 2/4 has no medium beat, so four quarters would be most commonly emphasized like this:
ONE two ONE two
This emphasis gives a noticeably different feel to the music than 4/4 does.
It also has to do with strong and weak beats. In 2/4, everyother beat (the one in each measure) is a strong down beat. In 4/4, every fourth beat (the one in each measure) is a strong down beat. It is all about accents and feel, so there is a major difference.
It's all about the feel in the music. Sometimes it really doesn't matter, and it is very difficult to say what is more natural.
But often you have distinguised beat at the first in each measure.
If it is a march, you have the first beat on the right foot when marching to it, and it is natural to have a 2/4.
Rock music often has a beat pattern that more or less repeats itself each 4th beat, and thus it is more natural to have a 4/4.
@MatthewRead's answer above hits the nail on the head. There is another reason that 4/4 may be used instead of 2/4 and that's for the ease of reading the notes from the page.
For example, in American Oldtime music there is a strong 2/4 feeling to it, but it's easier to read if no note values are below an eighth note or a dotted eighth. So the music is notated as four beats to the bar (measure) even though the music is played two beats to the bar: half-notes are notated as whole notes, eighth notes as quarter notes, etc.
Nowadays, 2/4 in traditional dance music is usually notated in 4/4, although older books often notate it "correctly" in 2/4 and if you can't handle a dotted sixteenth may God have mercy on your soul.
Sorry, but "one-two, one-two" would be the way I would count 2 bars of 2/2 time.
I count 2/4 time as "ONE-and-TWO-and". The beat is bouncy in motion and "rocks" back and forth from one count to the next. It creates a firm downbeat and backbeat .
I count 4/4 time as "ONE-two-three-four". The beat is circular in motion and "rolls" from one count to the next. You can play faster with 4/4 time.
A time signature is somewhat arbitrary... each player is free to express himself or herself in whatever time frame works best for him or her. Fiddlers often write out their music in 2/2 time (to reduce repetitive eighth note passages) but actually play in 2/4 time.
I teach beginner acoustic guitar and play bluegrass, most often expressed in 4/4 time -- and old time music, most often expressed in 2/4 time. We play regularly in 2/4, 4/4, 3/4 and 6/8 times.
The short answer: The difference is whether it's intended that we emphasize 2 or 4 beats per measure, and this distinction is ultimately pretty arbitrary.
Many answers have given great explanations for the difference between 4/4 and 2/4. Here's a practical example, so you can actually hear what it sounds like. The time signature change is very, very obvious:
"Pink Label" by Laysha (YouTube link)
It abruptly switches from 4/4 to 2/4 at around timestamp 0:35, then switches back around 1:10. The same change happens again around 2:00.
Even if you don't know anything about time signatures or rhythm, you should immediately recognize that the song sounds "faster" at the transition, though the tempo stays the same. Changing which beats are accented can achieve this.