I am not an avid fan of football, but whenever I happen to watch a game that goes to extra time, the score doesn't change, and the game has to be decided by a shootout.
How often does extra time actually change the outcome of a game?
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Data, data, and data :-)
Went manually through the history of two major tournaments, the World Cup (main tournament only) and the European Championship (through round of 16 games in 2016). It's probably reasonably fair to call these the two predominant national team cups to use extra time (Copa America has apparently typically gone straight to penalties except in the final?). Wikipedia was used as the source. While not the cleanest source, I've found it to be fairly reliable on such data, and the format facilitates looking through it quite well.
In those two tournaments, out of 82 matchups to go to extra time, 38 (46%) were decided in extra time.
However, counting games from the short with the golden goal policy (where the game would be automatically over when a team scored the first extra time goal, rather than finishing 120 minutes as is typical), is perhaps troubled (since it still could've still reverted to PKs with an opposing goal)... so a further breakdown:
(There were plenty of cries about the remote possibility of a coin toss in the World Cup group stages a couple years ago. And likewise when it is used to choose the home team for the rare single game MLB playoff qualifier. But to choose who advances to the final?!? And they call PKs a coin toss! Also happened to notice... it just happened in the 2015 African Cup... where the second team to move on from Group D, after tying across the board, was indeed decided in this strange drawing of lots.)
Certainly surprising to me to see such parity between the two possibilities.
Additionally, 71 goals have been scored in those extra time games, 0.87 goals per extra time on average.
And to make sure it's not biased by older games, or that trends look to be changing:
A dropoff, but not a great one, especially considering these are the equally matched (weeds out badly matched blowouts). And to reapply back to your initial question, in those past (almost) two decades, 17 of the 40 extra time games (43%) were decided in ET. Though the sample size isn't amazing, and we also haven't looked at other competition levels, such as other national team tournaments, domestic team cups, or continental club competitions... it seems rather reasonable to at least estimate between about 30-60% of extra time matches are decided in the extra time period.
Among those... the past two World Cup finals.
I don't know exactly "how often" the extra time will decide the game. But it does happen. A few prominent examples would be the Euro Cup Finals of 1996, because it was decided by a so called "golden goal" (with that rule in place, the match ends when one team scores a goal in extra time): UEFA Euro 1996 Final. Another very recent example would be the WC final of 2014.
The most recent example I can think of is the match between Croatia and Portugal in the round of 16 at the Euro 2016, which was decided in the 117th minute of the game.
But you are right, penalty shootouts are very common. I guess it's because no team want's to risk too much during extra time, because if you make a mistake you have very little time to correct it.
As to why "extra time"/"overtime" was invented: Not too long ago a match would have to be replayed if no winner was found after 90 minutes. If a knockout match ended in a draw a rematch was scheduled a few days later. This led to various matches that had to be replayed more than a few times. Of course in today's tight schedule there often is no time for a rematch, so the "extra time" was invented.
A penalty shootout was invented in the late 1970's. Before that, the winner of a match after a draw was decided by a coin flip.
In your question you are assuming that overtime/extra time does not change the score. Therefore I will focus on the second part of your question about shootouts
The shootout score is separate from the score from the game in regulation/extra time.
In some situations the winner of the shootout is the winner of the game, in other situation the match is still considered a draw.
Scoring and importance of the shootout victor depend on the league. For example:
"In the calculation of UEFA coefficients, shoot-outs are ignored for club coefficients but not national team coefficients"
"In the FIFA World Rankings, the base value of a win is three points; a win on penalties is two; a draw and a loss on penalties are one; a loss is zero"