Both stanch and staunch come from the Anglo-French estancher, meaning “to check or stop the flowing of.” Both have been in use for many hundreds of years. And most dictionaries will list them as having the exact same meaning. They are, in fact, variants of each other. But there's a catch: staunch is more commonly used as an adjective (it has several meanings in this role, including “steadfast in loyalty or principle” and "substantial"), and stanch is more commonly used as a verb (common meanings are "to check or stop the flowing of" and "to stop or check in its course"). Here are example of each in typical use:
a staunch supporter/advocate
staunch resistance/allegiance
to stanch the flow/bleeding
stanching the loss of jobs/revenue
Note that saying that something is more commonly used in some way does not necessarily mean that people who choose to use it in the less common way are wrong. There is a considerable body of evidence, from reputable sources, of staunch and stanch being used in their less common roles.
Some people will tell you that you should always keep these words apart, and if you’d like to do this you may find the following sentence of some assistance in helping you to remember the difference: "A staunch friend would help you stanch a bleeding thumb."
Alternatively, you may rely on the time-honored method of people-who-remember-things-poorly and use this limerick:
Tho’ neither stanch nor staunch must conform
To rigid semantical norm
Some editors will blanch,
When encountering stanch
If it’s used in adjective form
