Why are there such things as strong and weak beats? And how can we use them or consider them while composing a new music piece?
A lot of music - but not all, by any means - has a pulse running through it. The component which often gets the foot tapping. That is what usually dictates what the time sig. will be for a piece. When that piece is then written down, it's simple to divide the music into small sections which are designated bars or measures.
The first beat in most of those bars is the downbeat, and is quite often slightly emphasised, either on purpose or naturally, when being played. It happens in all time sigs, be it 4/4, 3/4, 5/4. 6/8 et al.
In 4/4, generally, there is another smaller emphasis on beat 3, sort of tying the whole bar tidily together. Same idea happens in 6/8, but can't in 3/4, due to its lopsidedness.
I think in a lot of cases, it's more a case of 'I've made up this tune, and gosh, it's in 4/4', rather than 'I want to make up a tune and it'll be in 4/4'. As in the timing is born intrinsically with the tune. It's not always the case: some tunes happen to have an odd bar with a different time sig., it just happens. And that could be an idea to use. Music should sometimes have the unexpected in it. Play a piece in 4/4, and suddenly there's a 5/4 bar!
According to my composition teacher, all time signatures are basically in groups of two or three . In groups of two, the first beat is strong and the second is weak. In groups of three, the first beat is strong and the second two are weak. In 4/4, since it is a multiple of two, the first and third beats are strong (though the third beat is less strong than the first) and the second and fourth beats are weak. The strong beats give a reference point which normally give the downbeat of a measure.
If we did not have naturally strong beats, there would be no syncopation. Syncopation places accent on beats that are normally weak (or portions of beats that are normally weak) and goes against our ears' usual expectation of strong/weak.
In terms of composing, you will hear in your head where you want the accents to be and determine whether those are in the regular strong places or if there is syncopation. Then you would choose a time signature or combination of time signatures to notate that. The music comes first, the notation later.
Examples: A walz is strictly heavy 1, light 2,3 . Rock music is heavy on 2 and 4 (in 4/4 time), and quite often the driving rhythm is eighth notes, played as medium-light; heavy-light, medium-light, heavy-light . Swing music in 4/4 may well "drag" the 4th beat so it shows up late, and the rhythm is heavy on 1, light on 2 and 3, and medium on 4.
In sum, the beat pattern helps define the type of music involved.