Well, I think the two are for completely different purposes. One does not have an "advantage" over the other. Like a pickup truck vs a roadster.
Enclosing a lanai (which we are planning to do) adds a room to the house,
These can be enclosed with floor to ceiling glass, so it has the advantage of being dry, and possibly airconditioned, and the advantage of feeling like you are "outside".
Others, like us, are looking to convert it into a part of the house, concrete block, etc, which typically is an airconditioned room that is dry and used for some indoor activity - game room, tv room, office, study, etc.
A birdcage is typically open to the elements. It is typically for "outdoor" activities - swimming, barbequing, sunning, lounging, etc. It typically keeps bugs and most leaves and animals out.
We are not outdoor people, so enclosing the lanai more fits our lifestyle and needsā„
Also, almost without exception, it is less expensive to make significant changes while the house is being constructed rather than after moving in. But, many do them anyway - like us.
If you have the room and money, I know some that have done both. Which solves any issues with making a choice
