I am not able to 'Add to cart' and update cart if I am logged out of my Magento store.
Could anyone give me some pointers on solving this?
The functionality is working on my main website but this is another store which I have created for the same website.
As Magento installation is the same, both the stores are using the same files.
There're two things you should check first:
b) the button doesn't have required classes attached
Oh, by the way, you can try turning off JS merge as this may cause the rest of (otherwise working) scripts to fail when one of the scripts throws an error.