In this tablature of the song I don't want to miss a thing it is notated with 2/4 time signature, but as one quarternote is 60bpm the way it is notated, essentially in eighth notes, it is like a standard 4/4 time signature with 120bpm. So why write it as a 2/4 with 60bpm when you could equally write it as a 4/4 with 120bpm?
Actually, you're looking at time signatures of 2/4 and then 4/8. If you played it with a time signature with 4 in the denominator at 120bpm, it would be twice as fast as at 60bpm. In order for it to sound the same, doubling the tempo (from 60 to 120) you'd have to halve the denominator. So really we're looking at 2/4 versus 4/8.
But back to your question: technically both are correct. Or, perhaps I should say that neither one is incorrect.
The distinction really comes down to tempo. When I hear this song, I think of it being more on the slower side of things, not on the faster side of things. As such, I intuitively want to think of this song more as 60bpm than 120bpm.
Mathematically, both will sound the same, with the same ratios among rhythms. But musically, a performer playing this conceptualizing it at 60bpm will give off a different energy/vibe than a performer conceptualizing it at 120bpm.