For context, I was a heavy player of Gal Civ II, and also played Gal Civ III for a while. I actually didn't have any build up of Gal Civ IV, just suddenly heard it was out (a great surprise for me). I have played about 150 turns so far, so this is my initial impression. I may add a few more notes as I continue my current playthrough.

Good

• In general it feels a simplier and more streamlined version of Gal Civ 3, which in general I have liked. Less "fiddlelyness" and more concrete action.
• I am enjoying the new leader concept, picking my ministers and my governors seems a pretty fun task, and the attributes make sense and our straightforward.
• The biggest improvement is in ideology. Ideology has honestly always been the weakest part of Gal Civ, with bonuses that have often seemed arbitrary, with points that are mostly generated through one time expansion bonuses. The new system seems enormously better, the initial planet decisions are more based on what I want to do with the planet rather than what ideology I want. I can make and match the ideologies more smoothly, and there are a lot more to choose from. I'm fully on board.
• I really like the new slot system over the old mass based one, though I am sure this will be controversial for the old school purists. Ships are easier and more concrete, with each slot feeling like an important part of the ship.
• I really like the new commander system. Its a great way to get the special ships of the old school Galactic Bazaar in a more thematic way.
• The new leveling system is fun and creates some neat ships. That said, the +20 hp/+20% hitpoint module seems pretty darn strong, but it does have a decent cost.
• The new galactic bazaar allowing me to buy all of the core resources is a big improvement, that was always a pet peeve of mine in Gal Civ 3.
• I think the expansion/economy is probably the biggest improvement in the game. The old game was rush rush rush, expand expand expand....more planets was just too powerful not too. In the new game, as most planets aren't all that great, and population is no longer a major factor in your resources (rather approval seems to be the key now for great yields).... I am not finding colony rushes the automatic default anymore. In general the economy bonus are much easier to follow (though I still find the approval calculation a mystery).
• The new policy system is pretty fun, not sure yet on the balance, but I find I get enough slots within a short time to do some mix and match combos and have changed things around periodically when the needs arise.

Bad

• The citizen attributes I very quickly lost interest in when assigning for scientists and workers and the like. It looks like the attributes aren't that important for those, but honestly I could care less.
• There is some UI clunkiness that is quite annoying. Switching buildings in a build queue, sometimes I can slide the build and have it stick, other times it won't, leading to frustration. I haven't found any way to remove buildings from a queue other than the delete button. 
• I wish their was a "fortify" button, sentry is decent but there are ships that I simply do not want to move period, and I wish there was a button for that.
• While I like the concept of colonies, I think its weird that the only difference is a governor. With a governor my planet cannot be attacked in the early game and suddenly doesn't have decay, and then I can knock it back to a colony if I want. Do colonies actually build buildings? What happens when I drop a governor, are my selections for buildings voided, or does it finish those and then continue with something else? Its very unclear.
• I'm finding it hard to click on individual ships sometimes, and I have to zoom in much more than I feel I should to make selections at times.
• I may change my mind as the game goes longer, but in general I find the cost of leaders too much to stick them in the faction areas, I always have use for my money elsewhere. For ministers and governors (and maybe diplomats) sure, but not for the factions.
• I generally find that AIs are far too demanding in trading even under normal difficulty. Equal deals should be the default unless the civ hates me, but normally I have to pay much more.
• I like the new control concept, but at least in the early game I find the cooldown on control powers much longer than the control points. I don't see the new to get more control points, as it takes forever on those cooldowns.
• With asteroids, I have not noticed the distance on them mattering at all. Its also very tedious to click on every asteroid and set it to the planet I want, as often they will send to colony worlds which is the last thing I want.
• The adjacency bonuses seems pretty arbitrary and mostly pretty weak. Basic manufacturing gets little bonus, but research labs get double the adjacency. Sometimes big buildings gets some sweet bonuses (like housing), but other big buildings get almost nothing.
• It took me a while to figure out that not all of my buildings are in the bottom left corner, and I can in fact build all my basic ones from the tile itself.
• I don't like that the basic buildings don't show their adjacency level by default anymore.
• I wish the basic starbase types were more distinctive. Mining starbases are just really good in general, for example if military starbases had greater range now suddenly that might be a more interesting decision, or if only influence starbases actually had borders, etc.
• The fact that constructor modules take up more than one slot is fine, but I can't for the life of me see where that is noted, and I was very confused for 5 minutes as to why I couldn't put more modules on my ship.
• The Altarian and Terran colors are so similar I literally can't tell them apart.