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Magento shop is using PrototypeJS library which is registering $ for itself. jQuery library wants to do the same thing, that’s why in order to use jQuery in a website which uses PrototypeJS (or similar library) you need to put it into Compatibility Mode. How?
Instead of:

$(document).ready(function(){
$("div").hide();
});

you need to write your code this way:

var $j = jQuery.noConflict();
$j(document).ready(function(){
$j("div").hide();
});

So as you noted the difference is adding: var $j = jQuery.noConflict();
and using $j instead of $.

You can read more on this subject here: Using jQuery with Other Libraries