If we had used a different hypothesis, the p-values would have been different. For example, if we had tested whether the difference was greater than zero versus smaller than zero, we would have got a very big p-value. The test-statistic would still be the same (3.67), but the way we compute the p-value would be different. In this case, we would take the area to the left of the test statistic (we would be looking at values as small or smaller than the one we got):

t.test(data$Score2,data$Score1,conf.level = .95,alternative="less",paired=TRUE) 

The preceding code generates the following output for the t-test result. We don't reject the null hypothesis: