The most important difference is that bilayer graphene has a bandgap of up to 0.25 eV, that depends on the magnitude of the electrical field applied in the direction perpendicular to the plane. Therefore bilayer graphene is a semiconductor and has different use in electronics, and as a sensor. For example, you can obtain better transistors than with single layer graphene. For some papers investigating bilayer graphene used in transistors (and comparison with single-layer graphene), check 

Article Electric Field Control of Spin Rotation in Bilayer Graphene

Article On the Possibility of Tunable-Gap Bilayer Graphene FET

Article Ultralow-Voltage Bilayer Graphene Tunnel FET

Article Multiscale Modeling for Graphene-Based Nanoscale Transistors

Article BIlayer Graphene Transistors for Analog Electronics

Article A Semianalytical Model of Bilayer-Graphene Field-Effect Transistor

Bi-layer graphene , when used for sensor , exhibits high sensitivity, good repeatability ,stable  specificity and fast response.

There could be new physics. Moreover, bilayer graphene has a gap.

The most important difference is that bilayer graphene has a bandgap of up to 0.25 eV, that depends on the magnitude of the electrical field applied in the direction perpendicular to the plane. Therefore bilayer graphene is a semiconductor and has different use in electronics, and as a sensor. For example, you can obtain better transistors than with single layer graphene. For some papers investigating bilayer graphene used in transistors (and comparison with single-layer graphene), check 

Article Electric Field Control of Spin Rotation in Bilayer Graphene

Article On the Possibility of Tunable-Gap Bilayer Graphene FET

Article Ultralow-Voltage Bilayer Graphene Tunnel FET

Article Multiscale Modeling for Graphene-Based Nanoscale Transistors

Article BIlayer Graphene Transistors for Analog Electronics

Article A Semianalytical Model of Bilayer-Graphene Field-Effect Transistor

I think it depends on the type of Bio-sensor and the mechanism you want to recognize the object. For example if you use zigzag graphene ribbon since its conductance at Fermi energy is very sensitive to the interaction with environment you can sense the Bio-molecule rapidly but if you have different Bio-molecules you can not recognize them from each other and you need to use armchair ribbon instead of zigzag ribbon since it can have band gap energy. Therefore by using graphene sheet with no band gap you may sense the object rapidly but by using two layer graphene sheet with band gap you may recognize different objects from each other better.

Best Regards

H.Simchi