What exactly is the difference between the PPP ("purchasing power parity") and LOOP ("law of one price")? For PPP &amp; LOOP: e = EP/P* = 1
So both are valid if goods in the US and EU cost the same in $ for example.
But then, what exactly is the difference between both?
PPP uses the price level, the price of the consumer basket. The price level can be identical without all prices being identical. An example:

Consider a consumer basket with just two goods and assume the weight of both goods in the basket is one half. If $p_1 = p_2 = 4$ then the price of the consumer basket is
$$
P = \frac{1}{2} \cdot p_1 + \frac{1}{2} \cdot p_2 = 4.
$$
However you get the same price level if you have $p_1 = 3,  p_2 = 5$.

So you can have two countries with identical price levels but differing prices. The converse of course is not possible. Thus LOOP implies PPP but PPP does not imply LOOP.