Brythan has a good answer. In addition, another factor which makes the counting process take so long in the United States, compared to other places in the world, is the sheer complexity of our elections. By which I mean, the number of individual questions placed on the ballot.
In the most recent Finnish election I could find, the 2015 parliamentary election, the voting instructions I found described a really simple ballot: write the number of the candidate you're voting for on a piece of paper, fold it in half, and put it in the ballot box.
I'm not as familiar with Wisconsin, but I can describe the ballot I had for the US presidential election in San Francisco, California.
As well as voting for US president, I was voting for the following offices:
- US Senate (statewide)
- US House of Representatives (districts)
- California state Senate (districts)
- California state Assembly (districts
- San Francisco county council (called Board of Supervisors here) (districts)
- San Francisco Board of Education (school district, county-wide)
- San Francisco Community College Board (county-wide)
In addition, others in San Francisco were voting for a representative on the regional transit agency (BART) board, also district-based.
I pick out the offices which are district-based because each of them has a separately defined district, which may overlap and intersect in interesting ways. That's in part because they split the pie up in different ways - California has 53 districts for the US House of Representatives, 40 for the state Senate, and 80 for the state Assembly - and in part because the rules for drawing the districts are complex, taking into account not just population and existing political boundaries, but also racial composition (so as to prevent underrepresentation of minorities).
As a result, just the county of San Francisco - the geographically smallest county in California - had 38 different ballots, depending on where in the county you live. Each ballot counted towards a unique combination of the district-based offices I listed above.
And San Francisco is simpler than other places in the state, because the city and county governments are consolidated. Just across the bay, Alameda County, where Oakland is located, has 14 cities, as well as significant territory not in any city. Los Angeles County has 88 cities.
Getting all the votes counted and applied to the correct office is a highly complex process, made even more complex by the fact that elections are administered at the county level, with each county choosing its own voting technology. Most of the districts cross county boundaries, so the state government has to consolidate vote totals for those offices.
I also had the opportunity to vote on 24 county-wide, 1 regional, and 17 state-wide referenda, but that's small potatoes compared to the complexity of all the different district-based elections.