delete a.x vs a.x = undefined

They are not equivalent. The main difference is that setting

a.x = undefined

means that a.hasOwnProperty("x") will still return true, and therefore, it will still show up in a for in loop, and in Object.keys(). Whereas

delete a.x

means that a.hasOwnProperty("x") will return false

You can't tell if a property exists by testing

if (a.x === undefined)

If you are trying to determine if a property exists, you should always use

// If you want inherited properties
if ('x' in a)

// If you don't want inherited properties
if (a.hasOwnProperty('x'))

Following the prototype chain (mentioned by zzzzBov) Calling delete will allow it to go up the prototype chain, whereas setting the value to undefined will not look for the property in the chained prototypes

var obj = {
  x: "fromPrototype"
};
var extended = Object.create(obj);
extended.x = "overriding";
console.log(extended.x); // overriding
extended.x = undefined;
console.log(extended.x); // undefined
delete extended.x;
console.log(extended.x); // fromPrototype

Deleting Inherited Properties If the property you are trying to delete is inherited, delete will not affect it. That is, delete only deletes properties from the object itself, not inherited properties.

var obj = {x: "fromPrototype"};
var extended = Object.create(obj);
delete extended.x;
console.log(extended.x); // Still fromPrototype

Therefore, if you need to make sure an object's value will be undefined, delete will not work when the property is inherited, you will have to set (override) it to undefined in that case. Unless the place that is checking for it will use hasOwnProperty, but it likely wouldn't be safe to assume that everywhere that checks it will use hasOwnProperty

To paraphrase the question:

Are delete a.x and a.x = undefined equivalent?

No.

The former removes the key from the variable, the later sets the key with a value of undefined. This makes a difference when iterating over properties of objects, and when hasOwnProperty is used.

a = {
    x: true
};
a.x = undefined;
a.hasOwnProperty('x'); //true
delete a.x;
a.hasOwnProperty('x'); //false

Additionally, this will make a significant difference when the prototype chain is involved.

function Foo() {
    this.x = 'instance';
}
Foo.prototype = {
    x: 'prototype'
};
a = new Foo();
console.log(a.x); //'instance'

a.x = undefined;
console.log(a.x); //undefined

delete a.x;
console.log(a.x); //'prototype'

If a.x is a setter function, a.x = undefined will call the function whereas delete a.x will not call the function.