What is the difference between $el and el in Backbone.js views?

mu is too short is exactly right:

this.$el = $(this.el);

And it's easy to understand why, look at the view's _setElement function:

_setElement: function(el) {
  this.$el = el instanceof Backbone.$ ? el : Backbone.$(el);
  this.el = this.$el[0];
},

This ensures that the el property is always a DOM element, and that the $el property is always a jQuery object wrapping it. So the following is valid even though I used a jQuery object as the el option or property:

// Passing a jQuery object as the `el` option.
var myView = new Backbone.View({ el: $('.selector') });
// Using a jQuery object as the `el` View class property
var MyView = Backbone.View.extend({
    el:  $('.selector')
});

What is a cached jQuery object?

It's a jQuery object assigned to a variable for reuse purpose. It avoids the costly operation of finding the element through the DOM with something like $(selector) each time.

Here's an example:

render: function() {
    this.$el.html(this.template(/* ...snip... */));
    // this is caching a jQuery object
    this.$myCachedObject = this.$('.selector');
},

onExampleEvent: function(e) {
    // Then it avoids $('.selector') here and on any sub-sequent "example" events.
    this.$myCachedObject.toggleClass('example');
}

See an extensive answer I wrote to know more.

In short, el gives you access to HTML DOM elements, i.e you can refer and access them, whereas $el is jQuery wrapper around el.

$el not only provides access to particular DOM element, moreover it acts as a jQuery selector and you have privilege to use jQuery library functions like show(), hide(), etc on the particular DOM element.

lets say that you do this

var myel = this.el; // here what you have is the html element, 
                    //you will be able to access(read/modify) the html 
                    //properties of this element,

with this

var my$el = this.$el; // you will have the element but 
                      //with all of the functions that jQuery provides like,
                      //hide,show  etc, its the equivalent of $('#myel').show();
                      //$('#myel').hide(); so this.$el keeps a reference to your 
                      //element so you don't need to traverse the DOM to find the
                      // element every time you use it. with the performance benefits 
                      //that this implies.

one is the html element and the other is the jQuery object of the element.