What is the difference between a Mechanic and a Repairman?
The basic difference is that a mechanic can perform maintenance on any aircraft; a repairman can perform maintenance only on aircraft for their employer, or that they own.

The full details are in 14 CFR 65 Subpart D (mechanics) and Subpart E (repairmen). Here are the basic privileges for each one (emphasis mine):


  A certificated mechanic may perform or supervise the maintenance,
  preventive maintenance or alteration of an aircraft or appliance, or a
  part thereof, for which he is rated (but excluding major repairs to,
  and major alterations of, propellers, and any repair to, or alteration
  of, instruments), and may perform additional duties in accordance with
  §§65.85, 65.87, and 65.95.


And:


  A certificated repairman may perform or supervise the maintenance,
  preventive maintenance, or alteration of aircraft or aircraft
  components appropriate to the job for which the repairman was employed
  and certificated, but only in connection with duties for the
  certificate holder by whom the repairman was employed and recommended.


"Repairman" also applies to experimental and light sport aircraft owners. If you build your own aircraft you have repairman privileges to maintain it, but you can't use those privileges to work on other aircraft, per 65.104(b) (emphasis mine):


  The holder of a repairman certificate (experimental aircraft builder)
  may perform condition inspections on the aircraft constructed by the
  holder in accordance with the operating limitations of that aircraft.