Both teams have demonstrated they can achieve similar offensive production. Does defense become more important? Is there a unique strategy regarding three-point field goals, steals, assists or other changes?
This is certainly situation based upon fouls(team and individual), pace of play, opposing schemes(offensively and defensively), etc.
Defensively, the strategy is to become more aggressive and force the opposing players into bad positions. Players are more tired in overtime so their shots are not as crisp and as clean compared to regulation so defense become a little bit 'easier'. There are a limited number of possessions in overtime, so defensively, you are trying to force a turnover or the worst possible shot, typically with a hand in the face. Little lapses in overtime turn into win or lose situations.
Offensively, teams become a lot more patient. Less shots are forced and less drives are put into double teams. This is wear a superior ball handler takes games over or a sound offensive scheme becomes more important. There are limited number of elite ball handlers at every level so the system being run is important. I always find I would rather have my players run off of screens instead of dribble the ball in overtime, less chance for turnovers.
But in the end, as @edmastermind29 and Herm Edwards said: "You play to win the game."
It's a very interesting question. In other sports, tactics do seem to change in overtime periods, mostly because teams try to "play for the shootout" or are just happy to tie if that's a possibility. Other teams really try to win during the overtime period. Because basketball's overtime rules call for a series of five minute quarters that will go on into infinity as long as the teams keep being tied at the end of them, I don't think there would be any major change to tactics.