Generally speaking a checklist is an actual list of items to check, and a flow is a pattern of movement across the aircraft controls (switches, dials, etc.) which will accomplish some subset of the items on a checklist.
A flow should be essentially "muscle memory" for a pilot and performed by rote, where a checklist may be more detailed and include more specific guidance.
As you can probably tell flows and checklists are closely related (with flows usually being derived from checklists). The two are complimentary tools that are used together: A pilot will follow a flow, and then verify that the required items have been completed using the checklist.
In larger/more complicated aircraft it's not possible to fully memorize some of the checklists, so flows are used to accomplish the "red box" immediate action items before proceeding to the full checklist.
As a simple example, consider the engine power loss flow & checklist for a Piper Cherokee.
Engine Power Loss Flow (Red Arrow & Numbers)
- Mixture (Set appropriately - usually "Rich")
- Throttle (Set appropriately)
- Carb Heat / Alternate Air (On)
- Magnetos (Both)
- Electric Fuel Pump (On)
- Primer (In & Locked)
- Fuel Selector (On a tank that has fuel)
Manufacturer's Engine Power Loss Checklist (Blue Numbers)
- Fuel Selector - Switch to a tank containing fuel.
- Electric Fuel Pump - On
- Mixture - Rich
- Carb Heat - On
- Engine Gauges - Check for an indication of the cause of power loss
- Primer - Check Locked.
- If no fuel pressure is indicated check tank selector position to be sure it is on a tank containing fuel.
- Ignition Switch - "Left" then "Right" then back to "BOTH."
- Throttle and Mixture - Different settings.
(This may restore power if problem is too rich or too lean a mixture, or a partial fuel system restriction.)
- Try another fuel tank.
(Water in the fuel could take some time to be used up, and allowing the engine to windmill may restore power. If power loss is due to water, fuel pressure indications will be normal.)
In this example the flow accomplishes nearly all the checklist items, but it does so moving over the panel in one direction. As a practical matter the flow is accomplished by just moving your hand along the bottom of the panel: "Push the mixture. Push the throttle. Pull the carb heat. Twist the mag key. Flip all the switches UP (on). Pull the primer and make sure it doesn't move. Move the fuel selector lever to the other tank."
Completing the flow takes less than 10 seconds and can be done with one hand while your other hand is pitching for best glide speed.
The manufacturer's checklist for an engine power loss moves around the panel in a more haphazard way, but it also adds a few items (checking the engine gauges & fuel pressure indication), and provides more specific guidance on possible causes like fuel contamination (water in the fuel) & how to deal with them. These items could be important if the memory items from the flow don't restore engine power, assuming you have time (altitude) to spend troubleshooting them.