Should templated classes be precisely as fast as untemplated classes?

I have a non-general-purpose object that only exists to be in collections. A library I'm writing that defines the object, let's call it Collected, and two implementations of the collection, that should have advantages in different scenarios. Let's call them Collection1 and Collection2. The collection is central to high-speed processing in this application so let's say performance is paramount.

The code for Collected is identical between the two, except 1) the ctor param that takes its collection, and 2) the member variable storing it.

So, I'm tempted to replace Collected with TemplatedCollected<class T> with the obvious code change. Apps using the library could decide centrally which they want, and either define MyCollected as TemplatedCollected<Collection1> or TemplatedCollected<Collection2>.

However if there is any chance it is any faster, I could make a Collected1 and Collected2 class that are identical except which collection they support. (Again, the app could define MyCollected to their needs.)