A word or phrase denotes its literal meaning (i.e., its dictionary definition), and it connotes all the meanings and associations it bears in addition to its literal meaning. By extension, denotations are the literal meanings of words and phrases, and connotations are the meanings associated with them. The words childish and childlike, for instance, have the same denotation—both mean of or resembling a child—but childish tends to have negative connotations because we use it to describe bad things, while childlike tends to have positive connotations because we use it to describe good things.
Examples
Here are a few examples of denote used well:
My new companion Baccouch—the name denotes someone who’s mute—had unsettling body odour and an innocent, if occasionally unco-operative, temperament. [Independent]
The word “trial” must be used in quote marks to denote the absurdity of the proceedings at Moscow’s Khamovnichesky Court. [Wall Street Journal]
Even more radical was Mr. Vignelli’s use of gray, not green, to denote Central Park, and beige, not blue, to shade New York’s storied waterways. [NY Times]
Connote is often used where denote would make more sense—for example:
“Whereas smuggling connotes illegal activity carried out clandestinely.” [quoted in The Jerusalem Post]
A jumpstart connotes a boost of power received from some more powerful engine. [Source now offline]
Smuggling is by definition illegal and thus denotes illegality, and jumpstart literally signifies a boost of power, so denote would be more logical in both cases.
These are positive examples:
In the context of a supervisory relationship, advocacy of discriminatory ideas can connote an implicit threat of discriminatory treatment and could therefore amount to intentional discrimination. [Huffington Post]
[T]he words “women’s writing” connote not simply a literature made by women but one that arises out of, and is shaped by, a set of specifically female conditions. [Guardian]