What is the difference between "reaction in both directions" and "equilibrium"?
As far as I know, they are used indistinguishably and interchangeably. 
One or another could be chosen to emphasize whether you are talking about chemical equilibrium or just trying to show that a reaction can take place in both directions depending on the conditions (which can be understood as completely the same...).

I saw the $⇄$ symbol used to specify some conditions for the reaction to take place in one direction and other kind of conditions for the reaction to occur in the other direction, and $⇌$ if conditions are the same for both reactions

For example:



First case is not an equilibrium, but it's true that the reaction can go both directions using the appropriated reagents in each case. But that could not be expressed using $⇌$. 

The second reaction is an equilibrium, both reactions are happening continuously under the same conditions. 
It also could be expressed using $⇄$.

That is the only difference I can tell. Like Martin pointed out, IUPAC does not give more hints about it, so the question cannot be answered for 100% sure.