Two simple harmonic motions are represented by the equation $y_1=0.1\sin (100\pi t+\pi /3)$ and $y_2=0.1\cos \pi t$. The phase difference of the velocity of particle $1$ w.r.t. the velocity of the particle $2$ is:

Displacement $y_1=0.1\sin (100\pi t+\pi /3)$.Velocity $v_1=\frac{d{y}_1}{dt}=0.1(100\pi )\cos (100\pi t+\pi /3)=10\pi \cos (100\pi t+\pi /3)$.Phase of $v_1$ (${\varphi }_1$) $=100\pi t+\pi /3$.Displacement $y_2=0.1\cos \pi t$.Velocity $v_2=\frac{d{y}_2}{dt}=-0.1(\pi )\sin \pi t=0.1\pi \cos (\pi t+\pi /2)$.Phase of $v_2$ (${\varphi }_2$) $=\pi t+\pi /2$.The "phase difference" usually implies evaluating at $t=0$ or comparing terms. However, note the frequency difference ($100\pi$ vs $\pi$). If we look at the initial phases:$\Delta \varphi =(\text{Phase of }{v}_1\text{ at }t=0)-(\text{Phase of }{v}_2\text{ at }t=0)$$\Delta \varphi =\pi /3-\pi /2=-\pi /6$.