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-5% technology cost is the same.
+5% discipline is the same
Japan +15% Infantry combat ability > Shimazu +10% Infantry combat ability
but Shimazu also has +10% Morale of armies and +10% Land fire damage.
Shimazu ideas will also make you more money with +10% National tax modifier and +10% Production efficiency. Japan has +20 Global settler increase which will be better for colonising, but it's not like that's a game changer when colonising without it.
So with Shimazu you have slightly more money and better armies, and with Japan you'll have slightly faster colonising and less unrest with less agressive expansion. I'd go with Shimazu then.
http://www.eu4wiki.com/Japan
http://www.eu4wiki.com/Shimazu
But like I said, they're pretty simmilar. It's not like choosing one or the other will make things much harder for you as a player either way.
+10% infantry combat ability; -1 yearly army tradition decay
+10% morale of armies
+15% manpower recovery speed
+10 production efficiency
+5% discipline
+10% land fire damage
+20% trade steering
-5% technology cost
+10% national tax modifier
Japan has:
-10% stability cost; +5% discipline
-2 national unrest
+20% national manpower modifier
-5% technology cost; +10% institution spread
+10% ship durability
+1 yearly prestige
+20 global settler increase
-15% aggressive expansion impact
+15% infantry combat ability
Japan will complete colonies quicker, have more manpower, can take more land without triggering coalitions (honestly, this looks more impressive then it actually is), have somewhat better ships, and be able to deal with unrest somewhat better than Shimazu all things being equal. Shimazu will have the better economy and their military ideas make them the Prussia of the East. Their +10% land fire damage will give them a major advantage over any enemy in battle from Tech 16 onwards.
TL;DR: Japan's ideas are very good (better than most in the East and I'd take them over most of the daimyo as well [even Uesugi]) but I don't see a reason to switch from Shimazu's ideas unless colonization speed is a major issue for you.
Thanks for taking the time to answer.