Reflow heats up the whole assembly (SMT). Wave solder is used for through hole boards. The solder is melted into a bath, the board passes over the top low enough that the leads/pads of the parts wick up solder. So really, the parts would all have to be mounted from the topside and the bulk of the heating is taking place on the bottom side (but is conducted through the leads into the part so really the whole process needs to be evaluated). I believe flow solder is the same as a wave soldering machine.
If have to do both to one board, the wave process is generally done first unless there are no SMT parts on the bottom side.

Additional differences between reflow and wave soldering.

In reflow soldering, the solder is applied as a 'solder paste' typically by using a stencil mask and a squeege. The solder paste contains both a solder flux and the solder in the form of minute solder balls. The solder paste operation is done BEFORE adding the components to the board. The board is then passed through a reflow oven at which time, the solder paste melts to complete the soldering of the components to the circuit board. The reflow oven will have several heating zones designed to preheat, activate the flux, reflow the solder, and cool the board. The oven is setup so the components receive as little thermal shock as possible, and spend only a few seconds above the solder melting temperature.

In wave soldering, the components are placed on the board. THEN the board is placed in the wave soldering machine. Typically the first step in the wave solder, the bottom of the board passes through a fluxing operation which either sprays flux on the bottom or passes the bottom of the board through a flux foam. After the flux, the board will usually pass a pre-heater to activite the flux and pre-heat the board for the soldering. Then, the bottom of the board is passed through a molten solder wave (looks like a smooth waterfall only consisting of molten solder). Afterwards, there will be a stage of cool-down.

Some details of the process vary by how expensive the oven or wave-solder machine is, the complexity of the circuit board involved, for what kind of volume the production line is intended for, and how 'mil or space qualified' the circuit board is.

Is it possible to reflow solder thru-hole devices?  Put the solder down via a stencil, add the components and then reflow?

Yes

It is even possible to have some surface mount devices on the bottom of a through-hole board and solder both in a wave solder. The surface mount devices need to be held in place with adhesive, the lead pitch of the devices needs to be large (like 0.050" SO type ICs), and the ICs need to oriented to minimize bridging and shadowing (body of IC deflecting solder from the leads in the wave).

The process for stencling solder paste to do through-hole in a surface mount process is referred to as "paste-in-hole". The stencil hole size, stencil thickness, and allowable lead-size-to-hole-diameter must be taken into account to make sure a reilable joint is formed.

I would question running bottom mounted SMT parts through a wave solder machine. If the parts stuck, they would likely be exposed to the heat to long and be damaged (remember these parts would be submerged in the solder until it came out the other end). Not good in my mind.

Marshell there is a such a thing as flow soldering. This is done by using a solder pot and pumping the solder up thru a nozzle plate. This is mostly referred to as solective soldering though but some folks call it flow soldering.