How to use SELECT DISTINCT and CONCAT in the same SQL statement

The problem are the other fields; DISTINCT applies to the whole result. Probably the best thing is to do to separate queries or populate 2 different arrays; if you ORDER BY name, you can remove duplicates by copying into the dest array only when the name changes.

Don't use DISTINCT, use group by:

$sql = "SELECT CONCAT(clients.studentFirstName, ' ', clients.studentLastName) AS name, appointments.location, appointments.subLocation, appointments.appointmentAddress1, appointments.appointmentAddress2, appointments.appointmentCity, appointments.appointmentState, appointments.appointmentZip, appointments.startTime, appointments.endTime, appointments.date, clients.school
                FROM appointments JOIN clients
                ON appointments.clientID = clients.clientID
                WHERE CONCAT(clients.studentFirstName, ' ', clients.studentLastName) = '".$roommate."' AND clients.school = '".$school."' group by CONCAT(clients.studentFirstName, ' ', clients.studentLastName);";

Also be careful about XSS in $school and $roomate if this accessible outside.