The ternary style is generally used to save space. Semantically, they are identical. I prefer to go with the full if/then/else syntax because I don't like to sacrifice readability - I'm old-school and I prefer my braces.

The full if/then/else format is used for pretty much everything. It's especially popular if you get into larger blocks of code in each branch, you have a muti-branched if/else tree, or multiple else/ifs in a long string.

The ternary operator is common when you're assigning a value to a variable based on a simple condition or you are making multiple decisions with very brief outcomes. The example you cite actually doesn't make sense, because the expression will evaluate to one of the two values without any extra logic.

Good ideas:

this > that ? alert(this) : alert(that);  //nice and short, little loss of meaning

if(expression)  //longer blocks but organized and can be grasped by humans
{
    //35 lines of code here
}
else if (something_else)
{
    //40 more lines here
}
else if (another_one)  /etc, etc
{
    ...

Less good:

this > that ? testFucntion() ? thirdFunction() ? imlost() : whathappuh() : lostinsyntax() : thisisprobablybrokennow() ? //I'm lost in my own (awful) example by now.
//Not complete... or for average humans to read.

if(this != that)  //Ternary would be done by now
{
    x = this;
}
else
}
    x = this + 2;
}

A really basic rule of thumb - can you understand the whole thing as well or better on one line? Ternary is OK. Otherwise expand it.