You COULD use them interchangeably and nobody would notice, and there is a LOT of overlap in their usage. But there is a difference. "Glimmer" suggests something that shines, sometimes brighter and sometimes dimmer. It stays in one place--it doesn't move. It doesn't wink out quickly or suddenly--that would be "flicker." It is an area of light, not a bright point. It is dim, maybe even hard to see. "Shimmer" suggests that a surface with a "sheen" with bright areas that move and change. A candle glimmers, reflections on a lake shimmer. "Through the shifting fog, I saw the light of a cabin window glimmering." "When Project Gutenberg began in July 1971, the Internet was just a glimmer." "He was still sinking, for the light became fainter and fainter until it was a mere glimmer. Then it began to grow and brighten, and he knew that he was rising toward the surface."--Ambrose Bierce, "An Occurence at Owl Creek Bridge." "Whiles all the night thro' fog-smoke white Glimmer'd the white moon-shine."--Coleridge, "Rime of the Ancient Mariner." "...he could see the shimmery flash of her dress as she moved about the porch..." "The golden fields that shimmer in the heat..." "Of whispering pine trees or the shimmering birch..." "Glimmer" is often used figuratively, "shimmer" is not. You cannot substitute "shimmer" in sentences like "I had a glimmer of an idea."

진구 , glimmer comes from within, shimmer reflects from the outside. Stars glimmer. Water shimmers.

There is no difference. They can be used interchangeably.

Their main difference is that "glimmer" is faint; shimmer is not. Both are "wavering lights". In my view they are lights of very different quality, and the two words should not be used interchangeably. Dan gas given very good examples. In none of his examples may one word replace the other.

Danniel is right about their interchangeability. There's also another usage for glimmer that you can see a lot like 'a glimmer of hope' or 'a glimmer of eagerness', etc.