[SalesForce] REndering Pageblock table based on condition

I would just use a controller property to control the visibility. You also need to use the rerender property for the commandButton. Something like this

VF Page

<apex:commandButton action="{!show}" value="Show" id="cb" rerender="myTable"/>

<apex:pageblocktable rendered="{!showTable}" value="{!Departments}" var="dept_ref" id="myTable">

     <apex:column value="{!dept_ref.Subject__c}"/>
     <apex:column value="{!dept_ref.Syllabus__c}"/>
     <apex:column value="{!dept_ref.Employees__c}" headervalue="Emp count"/>
     <apex:column headervalue="Name" value="{!dept_ref.Name}"/>

</apex:pageblocktable>

APEX

public class myController {

    public boolean showTable      {get;set;}

    public myController() {
         showTable = false;
    }

    public void show() {
         showTable = true;
    } 
}

EDIT Most importantly, set layout="none" on the outputPanel

Also...

If you remove all rendered="true" attributes, I think your page should work much better.

You don't need to specify this attribute unless you are referring to a variable from a controller or you specifically need to turn off an element.

Also, you could simplify your table delimiter items to:

<td>
    <apex:outputText value="{!lineNumber}" />
    <apex:variable var="lineNumber" value="{!lineNumber+1}" />
</td>

Or even:

<td>{!lineNumber}
    <apex:variable var="lineNumber" value="{!lineNumber+1}" />
</td>