What is the difference between the mouseover and mouseenter events?

I have always used the mouseover event, but while reading the jQuery documentation I found mouseenter. They seem to function exactly the same.

Is there a difference between the two, and if so when should I use them?
(Also applies for mouseout vs mouseleave).

You can try out the following example from the jQuery doc page. It's a nice little, interactive demo that makes it very clear and you can actually see for yourself.

var i = 0;
$("div.overout")
  .mouseover(function() {
    i += 1;
    $(this).find("span").text("mouse over x " + i);
  })
  .mouseout(function() {
    $(this).find("span").text("mouse out ");
  });

var n = 0;
$("div.enterleave")
  .mouseenter(function() {
    n += 1;
    $(this).find("span").text("mouse enter x " + n);
  })
  .mouseleave(function() {
    $(this).find("span").text("mouse leave");
  });

div.out {
  width: 40%;
  height: 120px;
  margin: 0 15px;
  background-color: #d6edfc;
  float: left;
}

div.in {
  width: 60%;
  height: 60%;
  background-color: #fc0;
  margin: 10px auto;
}

p {
  line-height: 1em;
  margin: 0;
  padding: 0;
}

<script src="https://ajax.googleapis.com/ajax/libs/jquery/2.1.1/jquery.min.js"></script>

<div class="out overout">
  <span>move your mouse</span>
  <div class="in">
  </div>
</div>

<div class="out enterleave">
  <span>move your mouse</span>
  <div class="in">
  </div>
</div>

In short, you'll notice that a mouse over event occurs on an element when you are over it - coming from either its child OR parent element, but a mouse enter event only occurs when the mouse moves from outside this element to this element.

Or as the mouseover() docs put it:

[.mouseover()] can cause many headaches due to event bubbling. For instance, when the mouse pointer moves over the Inner element in this example, a mouseover event will be sent to that, then trickle up to Outer. This can trigger our bound mouseover handler at inopportune times. See the discussion for .mouseenter() for a useful alternative.

Mouseenter and mouseleave do not react to event bubbling, while mouseover and mouseout do.

Here's an article that describes the behavior.