The short answer is: No.
Two statements being true means that relative to some 'situation', 'scenario', or 'world', both statements are true, while two statements being equivalent means that the two statements are true in the exact same situations.
For example, suppose that today I am wearing my red shirt and it is also raining outside. Then the statements 'I am wearing a red shirt' and 'It is raining outside' are both true today. But they are not equivalent, because obviously it is not always the case that exactly whenever I am wearing a red shirt it is raining outside.
First of all, by 'equivalence' I am using the notion of logical equivalence (for which we indeed typically use $\Leftrightarrow$), rather than 'material equivalence' (for which we typically use $\leftrightarrow$). You understand the difference, right?
Now, statements like 'I am wearing a red shirt' and 'it is raining' are clearly about contingent matters, for which it is easy to see that both being true is not the same as them being equivalent. But what about mathematics, where 'truth' is typically understood as 'necessary truth'? That is, aren't mathematical statements like '$1=1$' and '$\pi $ is irrational' always true? And, as such, whenever two mathematical statements are true, aren't they true in the exact same scenarios, namely every scenario?
Well, it turns out that even mathematical statements can be seen as contingent, namely contingent upon the axioms we are assuming. The angles of a triangle adding up to the sum of two right angles is only true in Euclidian geometry, for example.
Now, as it so happens, $1=1$ is typically seen as a 'deeper' truth yet, namely as a logical truth, and those are typically regarded as necessary truths: in every world, no matter how we interpret '$1$', it will be identical to itself. But '$\pi$ is irrational' is not seen as a logical truth, but 'only' as a mathematical truth.
So, as you can see, things get a good bit complicated, and philosophers of mathematics are still discussing what 'mathematical truth' really is, so what i just wrote in the longer answer is far from definite. But, I think my short answer should be sufficient for you to realize the basic distinction between two statements being true, and them being equivalent.