Inferno:
http://www.3dap.com/hlp/hosted/inferno/BWO:
http://freespace.volitionwatch.com/blackwater/Nope...two completely different campaigns. Just because the very basic idea is the same doesn't mean that they are the same campaigns or even related. Check out Machina Terra while your at it because it also has a similar outline:
http://terra.sourceforge.net/However, there are major differences between all three. Inferno has

of Shivans, big capital ships, and even the Ancients seemingly integrated into their campaign. Machina Terra is also slightly Shivan oriented but I don't want to give much away their either. Then there is BWO which is the project that I'm most involved in (and Kellan and I did create it).
BlackWater Operations is more character/politically oriented. You play as Commander Aidan Mallory of the 101st BlackWater Squadron. BlackWater Squadron is a GTVI squadron and they go and do the things that need to be done. Ultimately, you have to make a choice on which side you join. An Earth/GTVA war is only a portion of everything that will happen in BWO. Different elements, different focus. I personally feel that all three are quite different and well worth the time to check out when completed.
And if you did miss any plot point from FS2, Earth/Sol System was cut off from the rest of the Alliance at the end of FreeSpace 1. So all of these stories have a plot point that involves making contact.