Can a new chain significantly improve the riding experience? If yes - what else can? I am a casual commuter (in average 2 days per week, 30 km per day), on a generic bike which I bought a few years ago. 

Yesterday I took the opportunity of having the bike reviewed and the chain and back cassette was changed (the technician said I am lucky to have gone so far and I believe him).

Coming back home after the repair was a new experience: significantly faster and easier.

My first question is whether an old vs new chain can change the riding experience in a significant way. I am trying to weight this vs a placebo effect (though I felt the same way this morning).

If the answer is "yes", than I would be keen on changing/fixing other elements which would help me. I want to keep the bike and the breaks are good (this is something I take care of since my life directly depends on them). Are there other parts I should explore changing/improving?

Note: I do not want to turn my bike into another bike, I am happy with the basic parts as they are. It is rather some elements which need periodic change (chain in the example above) or simple fixing which I am trying to locate. 

In other words the V-breaks are great, even though changing to disk ones could be much better. The frame is heavy but never mind - I will not switch it to a carbon one. On the other hand if making sure that [something] is tight is important, I would make sure it is tightened. The whole question arose when I saw that enormous difference after just changing a chain (and some other basic adjustments I do not know about, but they were just adjustments)
The one substitution that will probably give you the most bang for your buck is better tires. High-quality tires roll with much less rolling resistance, have better grip, weigh less, and roll over small imperfections more easily. And if you're riding on tires with knobby treads, but only riding on the road, tires with minimal/no tread will ride much faster and smoother.

Keeping your drivetrain clean and properly lubed will also make a noticeable difference.

Making sure your bike fits you will make a big difference, and you may want to get new handlebar/stem/saddle to achieve a good fit.