Those of you who remember Rescue from Midnight Castle might say that there's not much there to analyze. You're right, there isn't, but when I watched the first episodes of FiM, it didn't take long for me to see the similarities. There are many ways I can handle this, but I think I'll just take it chronologically.

First off, we got an introduction to the setting. FiM uses the storybook recitation, and MC just gives us a minute or two of ponies having a normal day. This normal day includes running out of Dream Castle, roller skating, and general froliking about. The important thing I want to mention here is Ember.She is the most adorable little thing, a young earth pony with no cutie mark. She talks to Twilight about her future and Twilight says "You'll grow up to be your own special little pony." **Awwww** I imagine this is where the FiM idea of blank flanks and earning cutie marks comes from. However, in the original, this idea was never expanded upon and Ember, after the Pilot, was never heard from again.

Then, we've got the realization and the help. Both shows tell us what the danger is right away. Twilight Sparkle discovers it in a book, while the ponies of Dream Castle experience it first hand. The dragon-things attack immediately after 80s Twilight's talk with Ember.
While FiM Twilight comes to Ponyville and helps knowingly and willingly, Firefly steals Megan away to help, without the girl knowing what's going on. In FiM, the Ponyvillers are the ones who don't know the danger.

Once all of the Ponyvillers discover what is happening, another big similarity becomes clear: both villains have the same goal of everlasting night. So we have both groups going on a journey to find the MacGuffin that will defeat the villain. On both journeys there is a chasm and a river and a water-dwelling creature. The Twilight team finds their own strength, while the Megan team finds the Moochick and his 'Rainbow of Light'. Yup, the 80s ponies have relatively little talent, aside from unicorn magic and Firefly's 'double inside out loop' they are unskilled.... but they have a lot of courage, I guess.

The final fight brings, for me, the best comparisons and contrasts. In both versions, there is transformation and rainbow tornado. Scorpan, who I had a little girl crush on for some reason, transformed from a monster into a human prince (who is never heard from again) and all the guards and animals and transformed ponies are returned (presumably) to their original forms. In FiM, only Luna transformed, unless you count the mane 6 'Avatar State'.
Both the Elements of Harmony and the Rainbow of Light defeat the villain by surrounding it with colors, tornado style. I just love Rainbow Tornado.

So that's it. The first episodes of both series are almost the same thing, see? Only you don't need to take into account most of the events of Midnight Castle when watching the rest of the series. 80s MLP had little to no continuity: the same ponies were there, but the focus wasn't on a small group of friends but whichever random ponies they wanted the episode to be about. Pretty much the only constantly recurring elements are Megan, the Rainbow of Light, and Spike.