Is there any long-term advantage early treatment of congenital hypothyroidism?

Most Definitely! Congenital hypothyroidism is a medical emergency because postponing treatment may lead to mental retardation. The infant brain must have appropriate thyroid hormone levels for normal development. This is the reason we do newborn screening for this condition. The soonest we get treatment started, the better for the baby.
Yes. The rationale behind newborn screening is that early detection will make a difference. Early evaluation and treatment of hypothyroidism is key; untreated, I have seen estimates that about 50% of future IQ can be lost by six months. In addition, general growth other organ systems besides the brain and nervous system can be affected. So, yes, very meaningful advantages to early treatment.
Yes. The treatment need to start asap to improve brain development and maturation.
Yes. The term "goiterous cretins" was used for congenitally hypothyroid infants who made up half the population of state institutions in the late 1800's. Mental and physical delays marked their short life. Hypothyroid adults may tolerate periods without treatment, but a developing babies nervous system can suffer perminent damage without early treatment. This is why states pay to screen all newborns.

Related Questions

What are the long-term effects to early treatment of congenital hypothyroidism?

The earlier better. As early as possible (preferably right after a pediatric endocrinologist has had an opportunity to get a thyroid scan first). Iq points are lost the longer treatment is delayed.