variable === undefined vs. typeof variable === "undefined"

I'd stick to using typeof foo === "undefined" everywhere. That can never go wrong.

I imagine the reason why jQuery recommends the two different methods is that they define their own undefined variable within the function that jQuery code lives in, so within that function undefined is safe from tampering from outside. I would also imagine that someone somewhere has benchmarked the two different approaches and discovered that foo === undefined is faster and therefore decided it's the way to go. [UPDATE: as noted in the comments, the comparison with undefined is also slightly shorter, which could be a consideration.] However, the gain in practical situations will be utterly insignificant: this check will never, ever be any kind of bottleneck, and what you lose is significant: evaluating a property of a host object for comparison can throw an error whereas a typeof check never will.

For example, the following is used in IE for parsing XML:

var x = new ActiveXObject("Microsoft.XMLDOM");

To check whether it has a loadXML method safely:

typeof x.loadXML === "undefined"; // Returns false

On the other hand:

x.loadXML === undefined; // Throws an error

UPDATE

Another advantage of the typeof check that I forgot to mention was that it also works with undeclared variables, which the foo === undefined check does not, and in fact throws a ReferenceError. Thanks to @LinusKleen for reminding me. For example:

typeof someUndeclaredVariable; // "undefined"
someUndeclaredVariable === undefined; // throws a ReferenceError

Bottom line: always use the typeof check.

For undeclared variables, typeof foo will return the string literal "undefined", whereas the identity check foo === undefined would trigger the error "foo is not defined".

For local variables (which you know are declared somewhere), no such error would occur, hence the identity check.