No, all golf balls are not the same. At golf, the difference may not have a big impact on your performance until you reach a certain level. But it is key from the beginning when playing mini golf.
Believe it or not, the equipment that will influence the most your game is not the putter but… the ball!
There are differences from one ball to another, especially a compression rate which shows the hardness of the ball. It can have an impact on the way golfers play, especially for long shots (drives and irons), for putting the difference is probably less important.
Most courses use mainly (or exclusively) very soft balls, so soft that you usually don’t find it in regular golf courses, we usually call it “soft” balls in opposition to the “bouncy” balls (regular golf balls). As you figured, the main difference is the rebond (it will be quite low with a mini golf ball when you drop it, whereas a normal golf ball will have a much higher rebond). The main reason why mini golfs use non bouncy balls is that new players usually hit the ball way too hard. A harder, thus bouncy ball can more easily get out of the course; it gives a bit of tolerance as the ball will go way slower than a normal, bouncy ball.
Starting with a soft ball can be a good idea, but as soon as you feel a bit comfortable with your putt, go for the bouncy ones.
Depending on the hole you are playing, the ball will instantly impact your performance.
For example if the hole is winding and requests your ball to hit the border one or more times, the soft ball will lose a LOT of its speed at the first border hit, when the bouncy one will go way further. It may save you one or more strokes!
On the contrary, if there is a tunnel or any kind of obstacle on your way, if you don’t feel really confident to aim correctly, the soft ball can be a good choice. Obviously, the bouncy ball is going to “bounce” back much more if you hit the obstacle thus penalize you by coming back way further.
On the contrary, the soft ball is going to stop much closer to the obstacle and give you more chances to go though on next attempt, and if the ball goes through, the performance on a clear shot doesn’t really change from a ball to another so although it will no go as far as the bouncy ball the difference won’t be really significant.