This is a personal question on how your own strategy, positioning and go-to moves change when you're grappling with a gi vs. without one. Maybe they don't change at all.
For myself (non-scientific, just personal preference)...
Positions I like/use more with a gi than with no gi: guard, mount, back mount, north-south
Position I like/use less with a gi than with no gi: in guard, kasa gatame, knee on belly
Submissions I like/use more with a gi: chokes from the guard or mount, brother/sister series from the guard
Submissions I like/use less with a gi: knee bars, ankle locks, neck cranks largely because they are not allowed in some styles and omoplatas and far side armbars because there is "too much fabric" to effectively pivot without gettting caught or pinned down by your opponent
Takedowns I like/use more with a gi: any fall-away throw (duh) like a cement-mixer or double-overhook "falling rice bag" counter to a double-leg, inside leg sweeps
Takedowns I like/use less with a gi: duck-unders, armdrags and any type of takedown trying to work from behind an opponent, head snaps/leg shots in any sequence where you change levels
Despite that list, I'm neutral on more things than not, meaning I either use or discard them equally in both gi and no gi grappling.
The major difference in my game (besides the obvious Gi Chokes) is usually my approach to the clinch. With nogi I am more likely to control the head and arms. I am also alot more likey to do things that involve changing levels like duck unders and freestyle takedowns.
Once I get the gi on, things are alot more stalmateish. Most wrestling techniques are too easily stuffed by holding on to my gi. In return I usually control my opponent with his lapel on his chest, searching for an underhook to set up a trip with my other arm.
Come to think of it... I would love some judo lessons right now. Most of my gi sessions of late where we both started standing have been lessons in boredom.
Yeah...the gi changes everything. From just about any position it is easier to attack and harder to defend with a gi on. I feel that getting to a postion of advantage first (while wearing a gi) is important. My mentallity while wearing a gi is more urgent and aggressive to get on the offensive first and impose my will while keeping my opponent on the defense.
Without a gi on...it seems there is a little less to worry about and you can defend, sweep or escape a little easier. I also get tired quicker with a gi. I equate grappling witha gi to swimming in a weight vest.
I usualy try to get a guy worried about a gi choke while setting up something else.
While standing there are many more throws that you can actually pull off wearing a gi that you can't with out one. Having those handles makes all the difference.
The double overhook sacrifice throw that Fullerene describes is one that I like to use against wrestlers while in no gi. I have to keep my hips out and be quick with it though !! I rarely try it with a gi on.
I'm no judo expert, but I can pull off the basics with a gi on adecent percentage of the time. Osoto Gari, O Goshi and Ippon Seionagi are much harder without a gi. At least in my experience.
I don't think much changes for me, except that with no-gi, it's a lot faster paced.
I've never grappled with a gi. I have with shirts though so its similar using it to pull guard. Can't choke your opponent as easily with it though. I prefer not to use shirt/gi gaining any sort of advantage when grappling.