What is better, appending new elements via DOM functions, or appending strings with HTML tags?

I have seen a few different methods to add elements to the DOM. The most prevelent seem to be, for example, either

document.getElementById('foo').innerHTML ='<p>Here is a brand new paragraph!</p>';

newElement = document.createElement('p');
elementText = document.createTextNode('Here is a brand new parahraph!');
newElement.appendChild(elementText);
document.getElementById('foo').appendChild(newElement);

but I'm not sure of the advantages to doing either one. Is there a rule of thumb as to when one should be done over the other, or is one of these just flat out wrong?

Solution

Some notes:

Using innerHTML is faster in IE, but slower in chrome + firefox. Here's one benchmark showing this with a constantly varying set of <div>s + <p>s; here's a benchmark showing this for a constant, simple <table>.
On the other hand, the DOM methods are the traditional standard -- innerHTML is standardized in HTML5 -- and allow you to retain references to the newly created elements, so that you can modify them later.
Because innerHTML is fast (enough), concise, and easy to use, it's tempting to lean on it for every situation. But beware that using innerHTML detaches all existing DOM nodes from the document. Here's an example you can test on this page.

First, let's create a function that lets us test whether a node is on the page:
```
function contains(parent, descendant) {
    return Boolean(parent.compareDocumentPosition(descendant) & 16);
}
```
This will return true if parent contains descendant. Test it like this:
```
var p = document.getElementById("portalLink")
console.log(contains(document, p)); // true
document.body.innerHTML += "<p>It's clobberin' time!</p>";
console.log(contains(document, p)); // false
p = document.getElementById("portalLink")
console.log(contains(document, p)); // true
```
This will print:
```
true
false
true
```
It may not look like our use of innerHTML should have affected our reference to the portalLink element, but it does. It needs to be retrieved again for proper use.