Forgive me if a get a detail or two wrong, but it's been a while since I've played the original... (Also, I don't know anything about 5 beyond what I've heard; some of the changes they made to that one turned me off to it, honestly.)
The overarching difference: The original is, on the surface, much simpler than the ones to follow. Note though, that "simpler" does not always mean "easier."
There are many things you might be familiar with from previous games that are simply "not there" in Civ1:
-Culture (Introduced in Civ3; your civ's territory in 1 and 2 was simply whatever was inside your cities' radii)
-Hit points for units (Introduced in Civ2; combat is a one-for-all affair; winner comes out intact, loser dies)
-Stacks of Doom (Only became viable in Civ3 and after; combat mechanics in 1 and 2 meant that if a unit died on defense, all units in that square also died unless they were in a city or fortress)
-Workers (Introduced in Civ3; before then, Settlers performed those tasks in addition to founding cities)
Additionally, many concepts are extremely simplified compared to later games:
-Unit Experience (Until Civ3, units were either regular or veteran (+50% bonus to all combat). Civ3 changed it to experience = hit points, and Civ4 introduced a much more RPGesque experience system)
-Resources (Only modify production, trade and food output; until Civ3, no matter what resources you had, you could build any unit you had the tech for. Also, no special buildings were required to get the bonuses, although they did stack with terrain improvements.)
-Terrain improvements (Only roads, mines, railroads, irrigation, and forts in Civ1. Railroads give bonuses to production in squares that have them.)
-Diplomacy (Extremely simple in Civ1; you only had "Peace" and "War." There wasn't a transparent Relationships or Reputation mechanic, or an Open Borders agreement, or a permanent alliance. You could still trade techs, make demands, give gifts, and ask civs to attack one another, though.)
-Differences between Civs (Unique units and Leader bonuses were only introduced in Civ3. Before that, AI leaders had some personality, but it was along a continuum of "Aggressive vs. Pacifist," "Expansionist vs. Perfectionist," and one other axis that I'm forgetting; each could be either positive, negative, or neutral. Beyond that, every civ was the same out of the box except for the city name lists.)
There's a few other that don't come to mind at the moment, but I'll leave them to the experts.
The bottom line is: There's a lot of different mechanics that totally change what works as a strategy from game to game, so I can't say for certain that Civ1 is an easier game or not. By comparison, though, it's a much simpler one.