I would characterize "gusts" as a difference in measured wind velocity near the ground such as might be included in a METAR if the velocity exceeds 5 knots of difference with the average over the last minute or whatever the convention might be at the location where the observation is recorded.
"Windshear" could be defined in several ways depending upon the specific context in which the term is used:
1) A low level windshear warning may be issued at an airport if certain threshold values of wind velocity/direction/location differences are detected by airport LLWS detection systems. At so equipped airports in the US, the tower will generally issue the warning when the machine squawks and read off the recorded sensor differences. ( "North field wind 350/30G40, South field wind 270/20G30", etc.)
2) A windshear advisory may be issued due to a pilot report. ("A Falcon jet reported a 15 knot airspeed loss on short final for rwy 21 15 minutes ago.")
3) A GPWS windshear warning. The magic box compares data inputs according to it's programming and generates a warning. Reaction to such warnings vary according to circumstances and policy.
In a very real sense, windshears are the norm. It's just a matter of degree that we're concerned with. If the degree of shearing is considered to be great enough to constitute a threat according to established policy, a warning is issued.
Just as an aside:
Much of the impetus for the funding of such a warning capability as LLWS detection and the WS mode of GPWS can be traced to the DAL 191 crash in DFW. In that case, a microburst. It's one reason why they call the FAA "the tombstone agency!" Not fair perhaps, but then most major aviation legislation in the US can be traced back to singular events.