Every onchain transaction will require you to pay the same amount in fees. What will change between two different wallet softwares is the way how they calculate the "recommended fee".

Coinbase to Coinbase transactions are totally different since you are not doing any Bitcoin transaction, but only changing numbers inside Coinbase database. Every Bitcoin you deposit at Coinbase will give you a IOU that is worth what you deposited, and this is what you give to someone when you send them a transaction inside the Coinbase system. When you withdraw, they will remove those IOU from your account and send you real Bitcoins.