What are the differences between facilities field and maintenance field?

Facilities management is the bigger group. Maintenance management will only fall under the authority and responsibility of the facilities management.

the maintenance field is an importanat part of facilities field 

Facilities Management is the management of all the hard services in a company i.e. the buildings, factories etc. Ensuring that all buildings and related services (Plumbing, MEP) are operational and maintained within the required criteria to keep the business moving forward.

Maintenance Management is the actual monitoring and planning of the repair services of all equipment and hard services. 

A facilities manager will primarily deal with the "customer" The Maintenance manager will deal with the equipment (The Facilties manager would be the maintenance manager's customer)