○今日テークアウト食べようか (Let's eat takeout today)
×今日持ち帰り食べようか (Let's eat to go today)

○店員：お持ち帰りですか？ (Would you like it to go?)
×店員：テークアウトですか？ (Would you like it takeout?)
^-although you might hear this if they think you don't understand 持ち帰り, it's not "proper" Japanese

When I look up uses for them, the results I get indicate they can be the same? As labels (お持ち帰り/テイクアウト), or as requests お持ち帰りでお願いします/テイクアウトでお願いします or as questions to the customer お持ち帰りですか？/テイクアウトですか？

At any rate, I think I'll stick to the basic obvious uses till I get a better handle on it. Probably never even use テイクアウト.

Oh and also I get similar hits for 持ち帰り/テイクアウトにしましょう... I think the former is more common with volitionals?