How to use template module with different set of variables?

With Ansible 2.x you can use vars: with tasks.

Template test.j2:

mkdir -p {{myTemplateVariable}}

Playbook:

- template: src=test.j2 dest=/tmp/File1
  vars:
    myTemplateVariable: myDirName

- template: src=test.j2 dest=/tmp/File2
  vars:
    myTemplateVariable: myOtherDir

This will pass different myTemplateVariable values into test.j2.