I'd add that there are different types of scoring plays as JM talked about above. Each method awards pts due to degree of difficulty so to speak.
If you get the ball in the end zone by running or passing, its a Touchdown and you get 6 pts. If you can't reach the end zone, but get close, you can opt to try for a Field Goal, which earns you 3 pts.
On defense, if you tackle the ball carrier in the end zone, that's a scoring play called a Safety for which you receive 2 pts. Additionally, if the offense commits a penalty in the end zone, that's safety as well, resulting in 2 pts for the other team.
IF Touchdowns and Field Goals both resulted in 1 pt, it would take away any incentive for teams to forgo a Field Goal on 4th down, trying for a Touch Down instead.