how to compare two elements in jquery

Every time you call the jQuery() function, a new object is created and returned. So even equality checks on the same selectors will fail.

<div id="a">test</div>

$('#a') == $('#a') // false

The resulting jQuery object contains an array of matching elements, which are basically native DOM objects like HTMLDivElement that always refer to the same object, so you should check those for equality using the array index as Darin suggested.

$('#a')[0] == $('#a')[0] // true

For the record, jQuery has an is() function for this:

a.is(b)

Note that a is already a jQuery instance.

a.is(b)

and to check if they are not equal use

!a.is(b)

as for

$b = $('#a')
....
$('#a')[0] == $b[0] // not always true

maybe class added to the element or removed from it after the first assignment

You could compare DOM elements. Remember that jQuery selectors return arrays which will never be equal in the sense of reference equality.

Assuming:

<div id="a" class="a"></div>

this:

$('div.a')[0] == $('div#a')[0]

returns true.