Dimmer switch with ceiling fan instead of remote
A brief taxonomy of fan controllers

Ceiling fan speed controllers come in a variety of flavors, but there are only a few general types in the market that provide more than On/Off control:


Receiverless wallbox -- your garden variety "Fan speed controller" that only controls fan speed.  All the control logic is found in the controller in the wallbox, and it expects to be wired to a fan only.  Controlling a light kit is done using a separate wire from the wallbox to the fan.
Receiver Wallbox -- high-end fan or fan/light controllers such as the Lutron Maestro.  These use a wallbox controller that communicates with a canopy-mounted receiver via the powerline between the wallbox and the fan.  They are in some ways the most flexible as they support multi-way control of both the fan and the light and also may support home automation, but handheld remotes are rare here.
Receiver Wireless -- your cheap "fan remote" that comes with a fan.  These also use a canopy-mounted receiver module, but it is purely RF-based using a proprietary RF protocol to communicate with the battery powered transmitter-remote.  Their cheapness means they don't support any fancy home automation functions or multi-way controls, and some only provide limited fan speed or light control compared to wallbox controllers.


...and why you shouldn't use a light dimmer to control a fan

Light dimmers are built to switch lights -- a lightbulb is a relatively benign load compared to a fan, at least as far as a dimmer's switching element (triac) is concerned.  This means that cheap dimmers are built using cheap triacs and minimal external snubbing, leaving them unprotected against accidental (premature) turn-on or worse due to what are called dI/dt and dV/dt effects.

Furthermore, a dimmer starts the lights at low coming from the off state.  This is fine for incandescent (or even CFL/LED) lightbulbs, as they don't care.  But, a motor that's managed through variable voltage needs to be brought up to speed after starting then slowed.

Fan speed controllers use more rugged triacs and extra snubber circuits to survive the dI/dt and dV/dt abuses imposed by driving a fan, and they also start the fan motor at maximum speed, then let you turn it down.