If they are different from each other, why would there be an association between them as well?
The important thing to keep in mind here is that a dependent samples t-test and a correlation are testing completely different things.
The correlation is measuring association between Task 1 and Task 2 scores for each participant. If someone scores high on Task 1, would you expect them to also score high on Task 2 (a positive correlation)? Or does their Task 1 performance not really tell you much about how they would do on Task 2 (a correlation close to 0)?
The dependent samples t-test is designed for situations where there is some correlation between measures --- the idea behind it is to control for differences across participants by looking at the difference between Task 1 and Task 2 for each participant. The stronger the correlation between Task 1 and Task 2, the less variation there will be in the difference scores. For example, if there's a strong positive correlation between Task 1 and Task 2, then someone with a high score on Task 1 will also probably have a high score on Task 2, someone with a low score on Task 1 will have a low score on Task 2, etc. The difference between Task 1 and Task 2 for each person will be pretty small. The dependent samples t-test is a one-sample t-test on those difference scores: It tests whether the difference between tasks for each participant is significantly different from zero. It doesn't tell you anything at all about an association between tasks (in fact, it's taking any association between tasks out of the picture by just looking at the difference scores). It also doesn't tell you whether Task 1 scores were overall higher or lower than Task 2 scores (i.e. what an independent samples t-test would tell you), only whether the difference between Task 1 and Task 2 for each participant is generally positive or negative (or not significantly different from zero).
Note that if there's no correlation between Task 1 and Task 2 scores, then the difference between Task 1 and Task 2 for each participant might be big or small --- there will be lots of variability. In that case, the dependent samples t-test will actually be less powerful than an independent samples t-test on the same data.
A paired dependent t-test shows there is a significant difference between task1 and task2 congruent and task1 and task2 incongruent. So does this mean the participants performance is due to the IV not due to just being good at the task?
Yes, that interpretation is consistent with those results. The fact that you see a significant positive correlation between tasks suggests that some participants are just better at these tasks in general than other participants. If someone scores high on one task, they're likely to score high on the other one as well (and vice versa). Your dependent samples t-test pulls out these overall participant effects by looking only a the difference between Task 1 and Task 2 rather than overall task scores. The fact that your dependent samples t-test is sig suggests that regardless of participants' raw scores, they tend to do a bit better on one task vs. the other.