What strategies and tactics are found in both and which ones are more common in the other?
Sensei’s Library has a pretty good discussion of this. Probes, shape, influence, miai, invasion, contact plays, light play, and group management are all strategies and tactics common to both board sizes.
i started enjoying 9-9 recently. compare to 19-19, there is more fight, less moyo. you start actions almost immediately after the game begins. i like that.
9x9 is great with an app like GoQuest when waiting e.g. at a bus stop. While not comparable with 19x19, it is still way more complex than it might seem on first glance. I play 9x9 games quite a lot and am always surprised that there is almost never any repetition.
I noticed that my relative strength seems higher on 9x9 than on 19x19; this is probably because I used to be a Chess player before taking up Go, and the 9x9 size does fit better into my brain module for board games than 19x19…
Edit: You can also argue, that 9x9 is a bit more similar to chess: one single mistake of medium size will most-likely lose the game, while on 19x19 it is not clear that a local death of a group always leads to an overall advantage.
@mark5000 also started a nice thread about 9x9 strategy
Highlighting a quote from that post: