No in your example, you did not use qoutes in referencing the name of the variable.
You use the variable reference Highscore as the name.
variable Highscore = 1000
And you do Highscore >= webstorage.LocalValue(Highscore)
That means your doing:
Highscore >= webstorage.LocalValue("1000")
Menaing you are referencing a variable called "1000" in the webstorage, while you should be refering to its name, getting you the value of the local stored "Highscore".
And yes, web storage .... is used for storing in the web browser ....