No, they are separate games, only the game play is similar, the first one is much more simple, and imo better. I love stacking thousands of little ships in the first one, and the second ones planet management gets annoying.

No, the storyline is pretty incidental.
Which is better, though?
MoO 1. Every game should be just as complicated as it needs to be and no more. The same holds for graphics, sounds, 3D, fancy control systems, physics, system requirements, etc.
MoO is a good, quick, simple game - it takes minutes to learn (or remember how to play) and instantly you're in the game. All the animations / cutscenes can be skipped, and the interface is just enough to do everything you would want. You can complete a "Huge" Easy map in way less than an hour. MoO2 was the same thing, prettied up, with some nondescript changes.

I think both are strong but I enjoyed MOO1 more, mainly for the reasons posted above. Simple planet management based on a few sliders, but you can manage your empire fine that way
MOO2 seemed to take a long time to develop planets and get the fun part, war, going..
both are good though

MOO2 is more complex, prettier, and has more developed features. It seems like it SHOULD be the better game, but... I don't know. I can't say no to the original MOO's personality.
But hey, they're both spectacular games. You've got 'em both, play 'em both!

I encountered MOO2 first, years ago. I enjoyed it a ton. When I gave the original a shot, though, it just seemed a bit lacking. So while my choice is obvious, my bias probably is, too.

Play whichever you like.
Just a note. GetDaved has excellent play throughs with entertaining commentary for both Master of Orion and Master of Orion 2 on YouTube. GetDaved seems to like them both. Watch a bit of each series and see what you think. Now that I think of it, he has two play throughs for MoO2, one with human and one with Klackon. He plays on Impossible difficulty.
The only bad thing about GetDaved's videos is that it is hard to stop watching and get to playing yourself.

GetDaved cheated at the second playthrough with Klackons.

Oh noes! I am horrified! GetDaved has turned to the Dark Side?