As I've stated in the opening post, host's version is 5.6.29 while Wamp's version is 5.6.25.
But that wasn't the issue.
The issue was bad programming.
It seems that Wamp is much more strict when it comes to syntax than hosting, and the programmer who made the theme was often using <? instead of <?php and also often without trailing space.
Once I've fixed those issues the theme started working in Wamp as well.