With examples from The X-Files, The Mists of Avalon, Caprica, Katherine Kurtz's Deryni universe, & Star Trek: The Next Generation.
There are many ways to use mantle in the English language, but only one use of mantel.
Both mantle and mantel can be used to describe "a construction framing the opening of a fireplace and usually covering part of the chimney breast in a more or less decorative manner". When referring specifically to the shelf above the fireplace, one can also use mantelshelf or mantelpiece.
"He's still too young to really get what we're doing, Monica," I say, but don't stop her from finishing her stuffing of the stockings hanging on the mantel. 
Mantle, on the other hand, has several other meanings where it can be used. Below are a few of the more commonly used options.
- A mantle is "a loose, sleeveless cloak or cape".
Viviane waited patiently as Morgaine fumbled to settle the ermine-lined mantle about her shoulders.
- Recalling times when a chieftain or lord literally wore a mantle of leadership, mantle can also be "an important role or responsibility that passes from one person to another".
As the second, "replacement" son, Bill Adama was thrust too soon into assuming his father's mantle.
- Another metaphoric use for mantle is as either a noun or a verb that means "something that covers, envelops, or conceals".
In the wee hours before matins as Kelson and Dhugal sat vigil, darkness mantled the chapel in blessed silence, as if God Himself approved of their mission.
- In geological terms, the mantle is "the portion of the earth, about 1800 miles (2900 km) thick, between the crust and the core".
"Captain, we need to beam those people up immediately." Even over the comm system, LaForge's voice took on a strained quality that they all felt. "The planet's core temperature is rising, and I'm not sure how much longer before the core penetrates through the mantle."
So the easiest way to remember the difference between mantle and mantel is that a mantel can only be the shelf and surroundings of the fireplace. Every other use is mantle.
 Modified example from Inamorata: Chapter 24: Saint Nicholas Day by ariestess