Comparing electricity to the Internet is actually simple. Electricity changed things. It changed how humans lived life, worked, played, communicated, and so much more. The Internet did a lot of the same. It changed how we communicate, spend our free time, earn a living, and so much more (banking, investing, booking vacations, buying groceries, etc). The common denominator between the two: they brought about a cultural shift.
To ask the question "Which one is more important" is a little trickier, but only at first glance. While there are probably alternative ways to have an internet connection powered (solar, etc), electricity is probably the main way our modems and routers stay connected to the Internet. When our power goes out, the first thing I think, seriously, is "crap, no more WiFi." Now, if I had a generator, that might solve that problem, but I don't, and you probably don't either.
My vote for which one is more important: electricity.
Also, if I had to choose one to do without, I would choose the internet. As long as I have electricity I can have light to: write letters, pay bills, read (for leisure and education), watch public television (assuming if we don't have internet that we are a few decades in the past), heat and cool my home, etc.
Electricity is the chicken. Without it we have no egg (the internet) :-)
Thoughts? Which one is most important to you? Which one would you elect to do without?