As plants grow, cells replicate themselves. With that, they also replicate the guidebook for how to behave, the DNA. This is near-perfect. (un)fortunately, in the copying sometimes mistakes slip in. In many cases, this is not a problem and nothing happens. However, sometimes small mistakes cause big changes: The one branch that has yellow streaks in the foliage, witches brooms, ultra-short internodes. These mutations are the mechanism behind all the variation you see around you.
From that cell onwards, whenever it gets replicated, the trait is passed onto the next cell. So if this is a cell in the growing tip, the whole branch might have that trait and can be replicated by cuttings / grafting. If such a mutation takes place in a young seed, you get a plant with traits that differ from both the parent plants.
MOST variation in plants however are not only the genes, but also how they express themselves. Some traits are known to be variable by and from itself.
Take the Azalea example you mentioned. The flower color in Azalea is a very complex mechanism, which I do not know well except that with aging color shifts in certain groups of azaleas and flower color and pattern can switch between multiple versions. It is genetically embedded in Azalea to throw sports of a different color and/or pattern, yet considering the predictability I am not commilted to calling this a mutation.
Then if we go one step deeper, we see that plants adjust the whole growing habit to different climatic circumstances. From short internodes when growing in hot, dry and nutrient poor conditions, rankk tall weak stems under dark conditions allt he way to plants that have crawling stems in swampy marshes yet grow upright under drier conditions: That is a response to the environment, and pre-rogrammed. If you take a cutting, and change the conditions, the growing habit will change.