Takeout vs Pickup, is there a difference?

The words are used in different contexts.

When you call in an order by phone, it's clear that the food will be consumed offsite. However, the restaurant needs to know whether you will pick it up yourself (pickup) or you want them to deliver it to you (delivery).

When you place the order at the counter, delivery is not a possibility. In that situation, they need to know whether they should serve it on a plate so you can eat it in the restaurant (eat-in) or package it so you can take it out of the restaurant (takeout).

When explained this way, the origins of the terms become fairly obvious. They're just abbreviated forms of the actions that will be formed.