As a noun, grill means (1) a cooking surface using parallel metal bars, (2) a device that cooks with a grill, or (3) a restaurant offering grilled food. As a verb it means (1) to grill, or (2) to question relentlessly. Grille, which is only a noun, usually means a grating used as a screen or barrier on a window or on the front of an automobile.
Grille is often used instead of grill in restaurant names—e.g., Salt Creek Grille, O’Connell’s Irish Pub & Grille, Arooga’s Grille & Sports Bar. There’s no good reason for this. It’s just something some restaurateurs do.
To get all up in [one's] grill is to be extremely annoying, especially through nagging or by covering an opponent closely while playing a sport.
Also, in American slang, a grill is a plate molded to the teeth, usually decorated with diamonds or gold
Examples
These writers use grille correctly:
The 200 S has a black chrome grille and 18-inch wheels with black accents. [Automobile Magazine]
Police said the thieves climbed a security grille and broke through a second-storey window … [Stonnington Leader]
These writers use grill correctly:
Maryland lawmakers are planning to grill utility company representatives next week about why so many customers were left in the dark following last week’s snowstorm … [Washington Post]
It may be winter, but don’t try telling that to the grill — or the people who use them. [Delta Optimist]
And let’s not forget the American slang sense of grill:
Here Simms pulled off his gold grill, revealing plain white teeth. [Baltimore City Paper]

