Question

Differentiate Nusselt and Biot number amidst almost the same formula.

Answer 1

Formally both of them are indeed an dimensionless group h L / k, where h is the convective heat transfer coefficient between the wall of a solid body and an external flow, L is a characteristic length and k a thermal conductivity.
The difference lies in k :
In the case of the Biot number, k is the thermal conductivity of the solid.
In the case of the Nusselt number, k is the thermal conductivity of the fluid flowing around the body.
The Biot number will help you to know if your solid body can be considered to have an homogenous temperature (a "small body"). If Bi << 1 (typically Bi < 0.1) this will be the case. The difference between the surface temperature Tp and center temperature Tc will be very small compared to the difference between the surface temperature Tp and the bulk fluid temperature Tinf. Bi typically gives you the order of magnitude of (Tc-Tp)/(Tp-Tinf). To do such calculations, use L = volume/surface of the body. You typically calculate a Bi when you know h.
The Nusselt number will allow you to compute h from the characteristics of the flow. Dimensional analysis tells you that Nu=f(Re, Pr), Re Reynolds number of the flow, Pr Prandtl number of the fluid. You will find such correlations in heat transport books. Knowing Re and Pr, you compute Nu, and deduce the convective heat transfer coefficient h.