Significant difference and correlations Is it true that when there's no significant difference between groups then there will be correlations between groups?

My situation is as follows: I have a sample that was measured using two instruments $A$ and $B$. $A$ and $B$ are both questionnaires for measuring usability. 


*

*I compared data from $A$ with data from $B$ using a t-test to show that there's no significant difference between the means, because $A$ and $B$ are measuring the same thing. 

*I then used Pearson's correlation to see the correlation and found that they are correlated.


Can I do that? Can I say that there is no significant difference between $A$ and $B$, and interpret that to mean there is a correlation between $A$ and $B$?

One more question, how can I use a statistical test to justify that questionnaire $A$ is better than questionnaire $B$? The difference between them is that $A$ has constructs that weren't measured by $B$ and yet they have the same usability score for a software. Any ideas? 
 A: [Revised answer...]
OK, Now I think I understand it as follows: We have paired data, and the question has to do with the relationship between the results of the paired $t$ test and the correlation between the paired measurements.
They are not the same thing. You can have a significant correlation and a nonsignificant $t$ test, and vice versa. The $t$ test tests the difference between the means (or equivalently the mean of the differences), but the correlation coefficient measures a relationship independently of the two means.
But what is true is that a near-zero correlation means that the paired and un-paired $t$ tests will yield about the same result. With a strong positive correlation, the pairing is very advantageous and makes the paired $t$ a lot more powerful. Regardless, you should use a paired $t$ whenever you have paired data and an independent-samples $t$ when the data are no paired. (Provided of course the other underlying assumptions are satisfied.)
A: The OP urgently needs to clarify in what sense does he use the terms a) "non-significant differences" and b)"correlation" - but clarification a) is the most important.  
Example:
Assume we have two number series of equal length. Series $A$ is comprised of only $1$'s and series $B$ is comprised of only $2$'s. Is there, or is there not, a "significant difference" between the two?  
It depends on how we use the concept. We could argue that there is a significant difference, because the two series have no element in common.
But on another level, we could also argue that these are two very similar series because both have the characteristic that they are constant series of numbers (which is a very strong similarity trait, at certain levels of analysis).  
In short, the two series are totally different as regards their level, and at the same time they are totally similar as regards their structure.
