Nice in this context is senses 3 and 4, which relate to accuracy. The base meaning for these usages is "small" or "fine", so it means "precise" or "able to make precise observations."
We distinguish in order to separate or keep things apart which might otherwise be confounded
This means, We distinguish--i.e. make distinctions (probably with the implication of putting them into words)--in order to avoid mixing up things that could seem similar. Confounded, sense 1.
We discriminate with the further view of showing where in their differences consist.
Where you have "where in" I am assuming you meant "wherein."
"Discriminate" is used in sense 2, "to observe the differences between, distinguish accurately."
"With the further view of" = with the additional goal of
So the whole sentence means "Additionally, we observe the differences between things with the goal of showing precisely where they are different."
minute or tough
"Minute" here is pronounced "my-NEWT", with the emphasis on the second syllable. It's a different word from "minute" (as in 60 seconds). You want sense 3 of the linked definition, which means "small." The contrast is with "tough" which isn't a very exact opposite, though. I would have used "rough" instead.
Either way, the full sentence:
Distinction may be exact or not, minute or tough, broad or nice.
Can be paraphrased as:
The distinctions that we describe may be precise or vague, small differences or very large ones, in the broad outlines or in the fine details.
Hope this helps.