Based on my understanding, this test simply proves that the means are different, but doesn't estimate how different they tend to be.
A) Hypothesis tests prove nothing. They simply say that given the assumption that the two populations means are equal (plus all the other assumptions you make about homogeneity of variance and the distribution of the test statistic), it is incredibly improbable that we see a result at least this extreme or more extreme. Improbable things happen all the time. When we reject the null of a test, we essentially say that we prefer to believe we were wrong rather than believe we were lucky. But that is another conversation entirely. To the meat of your question now...
B) The p-value you get from the test does not tell you about the minimum difference, but a confidence interval can.
Let's start with an example in R. I'll simulate two samples with means 5 and 6 and then perform a t test.
pop1 = rnorm(25, 6, 0.75)
pop2 = rnorm(25, 5, 0.75)
t.test(pop1, pop2, var.equal = T)
Two Sample t-test
data: pop1 and pop2
t = 5.2148, df = 48, p-value = 3.864e-06
alternative hypothesis: true difference in means is not equal to 0
95 percent confidence interval:
mean of x mean of y
t.test function computes the difference between
pop2, so the confidence interval shown is for the difference is between
The smallest difference which is consistent with data at the 95% confidence level is a difference of 0.62 because it is the lower end of our confidence interval.