I have been thinking about these 2 hotels for a summer trip in Paris aswell. Here is what I have found so far:

Hotel Du Louvre is right by Musee Du Louvre, Jardin Des Tuileries and short 10 min walk to Musee de Orsay. It is also within 15 min of Notre Dame Cathedral.There is a metro stop right opposite to the hotel, so travelling from airport is very convenient. But you would have to change metro once in order to visit Eiffel Tower.

Now come to Park Hyatt Paris, it is at historical Place Vendome area. But a good 20-25 min walk to Louvre Museum and even further from Notre Dame Cathedral. Commuting to Eiffel Tower is however easier, as the metro there has a straight connection. If you are using Roissy Bus from Airport, then the stop at Opera is only few hundred metres from the Hotel.

After going through the reviews on the net and checking the hotel photos on the website; Vendome property looks more luxurious. It also has more restaurant options and a nice spa. Whereas, Hotel Du Louvre has lot of history and heritage behind it, the first 5 star hotel built in Paris 100+ years back, the place where Arthur Conan Doyle wrote few of his Sherlock Holmes stories and Sigmund Freud contemplated the mystery of Monalisa's smile.

Last thought, Paris-Vendome is already completely booked out for mid June and Hotel Du Louvre only has few rooms left. So hurry!