In the river section, does different paths change anything?

In the section where you are going down the river, you get to junctions where you can decide where you go.

Is there any difference between the paths? Any wrong paths compared to others?

Solution 1:

Per the wiki:

All paths will end up in the same scenario - with you fighting Ultros. However, there is one place that offers a great opportunity: about halfway through the river, you will be given the option of going Up or Left. If you choose Up, you will run in a loop until you return to this choice again. Many players use this as a training/grinding spot to gain a bunch of levels, since Bannon has such a powerful healing spell which doesn't cost any MP.