TDN is total dietary nutients or nutrition. Basically it's a matematical calculation of calories in the feed. Yes it is important, dairy cows need higher protien for milk production but they also need energy to convert the protien and stay in condition.
Quality hay is about 50-60% tdn, corn is roughly 80%. Dairy cows need 65-70%.
Timothy is lower quality hay than alfalfa, it has lower protien and tdn. Timothy left to head out becomes very poor quality hay... it is not bad if cut before it heads.