I see some people playing every note in a song on the guitar such as:
And some people only play the key notes:
When should you use each style?
Which style sounds best for someone who wants to play and sing at the same time?
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The second style is easier to play when you're accompanying yourself, as you only have to strum the chords. However, the more complex playing in the first example sounds better in a lot of songs, but it is harder to play like that whilst singing.Also, each song will ask for a different style of guitar playing, anyway.
First one is fingerpicking or hybrid picking with some arpeggios. Second one is chord strumming/rythm guitar.
If you plan to play alone and become a big act, like the great solo ones also do, I recommend you the first style, it is almost a must to fill the song on yourself. That is indeed the main function of fingerpicking, since it seems like another guitar is melodically joining.
It's more difficult but it is what makes a singer-songwriter (dylan, donovan, paul simon, silvio rodriguez, beatles, steve howe and many others), along with arpeggios.
On the other hand, if you only want to have fun in the park or play for a non ambitious audience without big career, it is muuuuch more easy the second approach.
Within historical context, fingerpicking has been used by folk artists, mainly singersongwriters with accoustic guitars, especially those in the 60's. So if you play the accoustic guitar and like to sound folky, no doubt it is convenient for you to make the extra effort! It's a basic resource along with arpeggios.
However, there are some caveats in fingerpicking over rythmic guitar.
Unless you really embrace it from your soul, fingerpicking is based on patterns, so it requires you to learn or invent a new one for every "groove". You have to be very brave to play compound signatures or concrete grooves/rythms. Traditional fingerpicking is especially rooted into 2/4 signatures with syncopated rythms. Arepggios can solve that more easily.
It's not as flexible for very fast changes
Chord/key changes are not as melodramatic as rythm, and cannot be stressed for a particular feeling or melodramatic effect so easily.