\(y = \frac{a}{2}\)

\( \Rightarrow \frac{a}{2} = a\sin (\omega \,t + {\phi _0})\)

\( \Rightarrow (\omega \,t + {\phi _0}) = \phi = \frac{\pi }{6}\) or \(\frac{{5\pi }}{6}\)
Physical meaning of \(\phi = \frac{\pi }{6}\) : Particle is at point \(P\) and it is going towards B
Physical meaning of \(\phi = \frac{{5\pi }}{6}\) : Particle is at point \(P\) and it is going towards \(O\)
So phase difference \(\Delta \phi = \frac{{5\pi }}{6} - \frac{\pi }{6} = \frac{{2\pi }}{3} = 120^\circ \)