得点 and 点数: scoring and totals
Distinguish earning points from recording a score, and additional points from the final total.
Earning points
得点, read とくてん, means scoring or points earned. 後半に二点得点しました。 says that two points were scored in the second half.
最初に得点したのは相手のチームでした。 says that the first score belonged to the opposing team. It does not determine the eventual winner.
Representing a score
点数, read てんすう, means a numerical score. 点数を表に書きました。 means recording the score in a table.
点数を間違えて書きました。 means writing the score incorrectly. It does not mean that the points actually earned changed.
An addition and a total
三点に二点を加えると、点数は五点になります。 says that adding two points to three makes five.
二点得点しました alone does not give a final score of two. There may be earlier points or later changes.
What is the total after adding two points to an existing three?
Show the answer and why
Five. Distinguish the two newly earned points from the total of five.