The Client field is normally used to indicate the entity that is paying for the work being performed on a matter. The Assignee is used to indicate the entity that has ownership of the matter (i.e. patent or trademark), provided the matter has been assigned. Often these entities will be the same. If so, only fill the Client field; if they are different, complete both fields. The Applicant field can be used to show who the applicants of a patent or trademark are, if different than the assignee or owner. 

One primary use of the Assignee field is when completing an AIA/14 Application Data Sheet (ADS) form for the matter, to be filed with the PTO. If the Assignee field is populated, that value will be used for the Assignee field in the AIA/14 form. Otherwise, data from the matter client field will be used for the Assignee field in the AIA/14.