It was initially planned as a general network authentication device for both public and private sector strong authentication needs. In 2009, however, the card was viewed by a government committee as a failure. There has been less than 300000 cards around by 2011 out of population of 5.3 million. The rationale to apply for a card has mostly been traveling abroad. Only few dozen government services have adopted it, and only one bank adopted it as login card to their netbank. All banks in Finland use a national standard called TUPAS, which uses one-time passwords. Banks also provide TUPAS authentication to other Internet-enabled businesses. Since TUPAS requires no dedicated hardware, cost of a card reader and card itself have been main causes in the failure of the eID card.
http://en.wikipedia.org/wiki/Finnish_identity_card
The problem is simply that smart card readers never got integrated into computers, and people didn't want to buy a USB dongle for that just to be able to authenticate with some government websites.
Instead, what happened was that the two-factor authentication system provided by banks became the dominant "secure authentication" method. By now it is supported by most sites that need such a thing, like banks, insurance companies, postal services, and several government sites.
I can for example authenticate with my bank user credentials to file my taxes (or could, when I was still living in Finland).
An additional benefit of the ID card is that you can use it for travel inside the European Economic Area. But I'll rather carry my passport with me, as that way I don't have to wonder whether the ID card is enough for my itinerary or not.
edit: ID cards are valid travel documents inside EEA (which is a larger area than Schengen)