Difference between logical equivalence and semantic entailment? When you have to prove that two formulas are logically equivalent, you make a truth table and check whether their their truth table columns are identical. And when you have to prove that a set of formulas logically entails a formula, you are doing the same thing. If I am correct, what is the difference?
 A: To show two formulas, $\varphi$ and $\psi$, are logically equivalent you need to construct a formal proof in the language of propositional logic of $\varphi \leftrightarrow \psi$.  To show that $\varphi \vDash \psi$ prove that whenever $\varphi$ is assigned the truth value, then $\psi$ also has the truth value, which as Mauro states is more related to $\varphi \to \psi$.  You can do this with truth tables. A priori, constructing a truth table is neither here nor there when it comes to logical (or syntactic) entailment.  A witness to logical entailment is a formal proof, and a truth table is not a formal proof.
Now, what we do have is the soundness and completeness theorems which say that $\varphi$ logically entails $\psi$ if and only if $\varphi$ semantically entails $\psi$.  These theorems, particularly the completeness theorem, imply that if you can show semantic entailment using truth tables then there exists a formal proof that will show logical entailment.  In fact, the completeness theorem is true constructively which is to say that not only does such a formal proof exist  in that case, but a constructive proof of completeness is an algorithm that will produce it.  The above theorems allow us to conflate logical and semantic entailment for propositional logic, but they are still different things.  If I ask you for a formal proof of $\varphi \to \psi$, constructing a truth table and then saying "by completeness" would be like if when I asked you for the inverse of a matrix you checked that the determinant is non-zero and gave me a program for inverting matrices with non-zero determinant.  What I expected was an actual matrix!  Of course, if I merely asked if the matrix was invertible, then the above would suffice. Luckily, this latter question is indeed what we are usually asking.
A: The most important difference between equivalence and entailment is that equivalence expresses a 'two-way' (symmetrical, if you want) relationship of logical implication, whereas entailment expresses this only 'one-way'.
Example:
$P \land Q$ is equivalent to $Q \land P$, since they imply each other
$P$ entails $P \lor Q$ (But $P \lor Q$ does not imply $P$)
Also, because entailment is one way, it can also be used to say that a set of sentences entails some othe sentence, which is another difference, since you would never say that some set of sentences is equivalent to some other single sentence.
