'Mix' means 'combine' : to mix, all you need to do is put different things together.

A bag of mixed nuts might contains peanuts, almonds and hazelnuts. A mixed class might have beginner and intermediate students in it. Sometimes you have to stir <em>in order to </em>mix together well (cake ingredients, for example), but sometimes you don't. Cocktails, for example - the best cocktails are shaken rather than stirred ( as all James Bond fans know).

'Stir' is a movement: for example, you put a spoon in a cup of liquid and move the spoon in circular movements so that the liquid moves.

It's possible to stir without mixing anything together: for example, if you have a cup of containing just coffee, you can stir it to to cool it down.

As verbs the difference between <em>stir</em> and <em>mix </em>i<em>s</em> that stir is to change the place of in any manner, to move; while mix is to stir two or more substances together. 

As nouns the difference between <em>stir </em>and <em>mix</em> is that stir is the act or result of stirring, agitation; while mix is the result of mixing two or more substances, a mixture.

One difference is when stirring, you would typically use a circular motion, whereas mixing need not be a circular motion.

Mixing doesn't require the different items to be indistinguishable. You mix a salad; you would expect the lettuce, tomatoes, and other ingredients to be distinct from one another.

Stirring would imply that the items become indistinguishable. When you stir sugar into coffee, you don't expect to see the two different ingredients; you expect to see one thing as the result.

I like to eat food in stir fry in a mix of variety of vegetable and meats. (In the context you may be easier to understand the distinction between mix and stir)

"Stirring" emphasizes circular movement.

"Mixing" emphasizes blending several different things together into one uniform thing.

Stirring a combination of things is <em>one way</em> to mix them.

Take a glass of water. Put a spoon in it. Move the spoon in a circle. You are stirring it. A vortex forms in the center. You might do it just because you like watching it. It starts out as pure, 100% water. It ends up as pure, 100% water. The water was "stirred," but nothing was "mixed."

Fill a paint can with half yellow paint and half blue paint. The colors stay separate and don't mix. Put a paint mixing stick into it and move it around in a circle. At first you see pretty spiral patterns of yellow and blue. As you keep stirring, eventually it gets thoroughly mixed. You have a can full of green paint. You mixed the paint by stirring it.

Fill a paint can with half yellow paint and half blue paint. Cap it and then put it into a machine like this <a href="https://youtu.be/mPWk8rx1C4w?t=11" target="_blank" style="background-color: rgb(255, 255, 255); color: rgb(17, 85, 204);">paint mixer</a>. Afterwards, you have a can full of green paint. The paint was mixed, but it was mixed by shaking, not by stirring. It was mixed, but not stirred.