The difference between a crevice and a crevasse is more than just a few letters. It’s the difference between geology and glaciology. While both terms come from the Anglo-French word crevace, to break, they mean two different things. Crevices are cracks or splits caused by a fracture of a rock, while a crevasse is a deep fracture in a glacier or ice sheet.
Crevasses form in the top layers of a moving glacier, usually because some parts of the massive body are moving at a different pace than the rest. If a glacier is moving over varied terrain (over mountains and down valleys), for instance, the glacier can stretch and fracture. The part of the glacier moving over a mountain will move more slowly, while the part that’s flowing down through the adjacent valley will gain speed. As the ice pulls apart at the stress points between those two portions of the ice, cracks form. Crevasses can also form when the glacier turns a corner, since the ice on the outside moves faster than the ice on the inside as it goes around a bend, or in open areas where ice begins to spread out horizontally (as it often does at its front end).
Often covered by snow, crevasses pose a great threat to mountaineers as they traverse the surface of a glacier: That snow can give way, leading to a steep fall. While the intense pressure at the bottom of a glacier typically squeezes a crevasse closed long before the crack reaches bedrock, the opening between the two parts of the ice can still reach as far as 100 feet down. As a safety precaution, climbers undergo crevasse rescue training in which they learn how to attach themselves to each other using a rope; should one of them fall into a crevasse, the other can pull them to safety on the ice.
A glacier may have more than one crevasse, and when they come together, they form a kind of free-floating column of glacial ice separate from the rest of the glacier. Called seracs, these pose yet another risk to climbers, because they can topple easily.
Meanwhile, a crevice doesn’t always pose a threat. These are formed when brittle rocks crack under stress, and the two sides begin to pull apart (rather than sliding past each other). While you wouldn’t want to fall down a large one, they’re not always dangerous—they can be as small as a crack you’d see in the sidewalk or as deep as a canyon. These rock formations are not typically a threat to hikers because they're typically easy to see; in fact, they can often be a great help to rock climbers. The small ones are great for hand-holds or to attach bolts to, and larger crevices can be scaled. They’re also habitats: In western North America, there are dozens of bat species that use crevices as roosts [PDF].
If you're wondering in the middle of a conversation whether to use "crevice" or "crevasse," it's probably safer to go with crevice—unless you spend a lot of time talking about glaciers. While it's technically a geological term, the word crevice is used much more generally to describe cracks and gaps, whether in rock or in other materials. For instance, engineers use the word to describe the gap between two joined metals. Weeds might grow in the crevices of a sidewalk or between bricks. Colloquially, you could describe the gaps in your couch cushions where all your change accumulates as crevices.
In short, if it's not ice and you aren't making a glacier metaphor, it's a crevice.