Since butter comes from cows milk, I am not sure how much the taste will differ from beef tallow. Beef tallow will have a mild beefy taste, which you might like or not.
The problem with using butter for grilling is that the milk solids will burn at the high temperatures normally used on the griddle. There are two ways to help that. One is to start the cook with a high smoke point oil (I use peanut oil, but anything other than olive oil should work). Then as you near the end of the cook baste the shrimp in butter. If you want to use a compound butter with herbs or aromatics like garlic, you can do so. I like adding rosemary to butter when basting.
Clarified butter is simply butter from which the milk solids have been removed. Thus, it does not burn like regular butter. Personally, I like the taste of regular butter better than clarified butter, but many would disagree. Use whichever you prefer.