Satin nickel is a plating. Brushed nickel uses tools to put small abrasions on the metal, all going the same direction on the metal. Both finishes hide water or dirt spots
better than polished nickel or chrome. Both finishes are comparable in
cost, and unlike chrome, may stay in fashion longer - or at least, not
fall out of fashion as quickly. Both are cared for in similar ways.

So will each kind look ok together in the same bathroom?

Generally speaking, brushed and satin nickel are very close in finish. If you think the items look OK together, then I'd keep them!

Very little difference between the two other than general make up. The brushed nickel may have a slight more sheen but would take a professional eye to catch it.

Brushed nickel has a warmer tone than satin nickel.