Train from Bruges to Rouen

These are not two train systems. They are listers of the trains and ticketing. They should both have the same trains offered.

You will have to change in Brussels and Paris, and in Paris this will involve crossing Paris from one station to another by métro. There are no other options. There are trains throughout the day, you can start as early or late as you like.
SNCB (Belgian Railways) runs the trains from Brugge to Brussels. SNCF (French Railways) runs the trains from Paris to Rouen. Brussels to Paris High Speed trains (TGV) are run by THALYS, which is jointly owned by SNCF and SNCB.
RailEurope is a ticketing agency that does not run any trains. DeutscheBahn is, as the name suggests, German Railways. They run trains in, to and from Germany, but they have the best timetable search engine on the web, so they are a good first call for finding train times.

As for buying the tickets, Brugge to Brussels has no discount for advance purchase, you can buy those on the day.
For Brussels to Rouen, you can buy those in advance for a discount at either SNCF or https://www.captaintrain.com/

Another option;
Bruges to Dunkerque
http://www.bonjourlafrance.com/trains_brugge_train_station_to_dunkerque_train_station_6949_80.aspx
Then Dunkerque to Rouen
http://www.logitravel.fr/trainssaleprocess/availability?anuncio=17649985&origenSV=Y1U&destinoSV=Y3U&idConsulta=GetAvailabilityFlightsWS@FE4E3247667323F70EBE5362ADED81E8&adultos=1&ninos=0&bebes=0&clase=&vuelosLowCost=on&residente=&acceso=ws&codMedio=749&divisa=EUR&timestampPET=14/04/2016%2023:03:19&timestampAGR=14/04/2016%2023:03:24&tipo=OW&fechaIdaSV=17/04/2016&ciaI=RLE&conI=ATR&numI=7318%2013109&agrI=&imp=57,20&tot=57,20&tax=0,0000&amc=aff.logitravel.31950.37711.7394&smc=43264158_kelbillet-sy_IT_M_search_result_offer__dunkerque_rouen

There are trains from Bruges to Brussels all day. You are probably only seeing ones in the late afternoon or evening because the RailEurope site does not show Belgian domestic trains, but only the small number of through Thalys international trains from Bruges to Paris. The DB site does seem to show the normal Bruges-Brussels services.

You may need to change the "clock" on the train website to an earlier time in order to see more train choices.

We just did a similar trip but trained to Lille right over the border and rented the car there. You have to switch trains somewhere at least once even on that shorter trip.

I suggest renting the car in Lille as the last poster did. That means a much cheaper train ticket, avoiding central Paris, and also may be faster, even though you drive more.

Should add that once we got on the highway in Lille the driving to Normandy was really easy. A lot of tolls but not anywhere near as much traffic as we're used to back home. And Lille is a nice place to visit as well, even if for just a little while, the old town is a short walk from the train station.

Blansief: Last fall, we took a local train from Bruges to Lille and picked up a rental car and drove to Bayeux--a very easy drive. Several cautionary notes:
1. There are two train stations in Lille: Gare Lille Flandres (local trains) and Gare Lille Europe (hub for TGV trains, Eurostar)--they are connected by a walkway of about 200-300 yards. We took a morning local train to Gare Lille Flandres (I seem to recall about 10-15 euro per person) and walked to the Gare Lille Europe to pick up the rental car.
2. The rental car offices are in a wing of the Gare Lille Europe main concourse and the cars are in an underground parking garage on the opposite side of the main concourse. My only complaint was that the GPS instructions were in French and we had to get the desk clerk to come change it to English for us--she was a bit miffed but when she had problems herself, she realized we had done the right thing. Also, they did not have a small paper map of the local area with which to orient one's self before departing.
3. The exit to the main autoroute south to Normandie is very convenient to the train station. I would recommend that you have the final address put into the GPS before departing. Also, familiarize yourself with google maps or similar ahead of time to understand the initial route onto the autoroute. I seem to recall that you initially follow the roads to Paris (in France, all roads lead to Paris--humor).
4. Please consider taking a break in Amiens and visiting the Cathedral--it is easy to get to from the autoroute and easy to return to the autoroute. The cathedral is one of the best in Europe. And, there are plenty of restaurants in the area for a meal.
5. Obey all speed limit signs.
6. There are plenty of rest stops with gas and food opportunities.
7. Our route took us over a neat suspension bridge near Honfleur--a great view and experience.
8. I highly recommend visiting the Bayeux tapestry while in the area--it is well worth it.
9. Remember, obey the speed limits.

Enjoy your trip and be safe.

~Darrel

We did a stop in Amiens as well for the cathedral and the Maison de Jules Verne (museum). Also stopped in a picturesque little town called Beuvron-en-Auge that someone recommended. And found an overlook in Honfleur where you get a view of the town and the bridge.