What's the difference between praying to Jesus and praying to the Father in the name of Jesus?
The difference is whom you're talking to. And I think it is both right and inevitable to pray to all members of the Trinity.
Having said that, I think it is normal, because of the way Jesus says it, to talk to the Father in the name of the Son. Because Jesus says, "Whatever you ask the Father in my name, I will give it to you."
That's the pattern that Jesus seemed to set up: "My Father is the great giver. I am the mediator between you and my Father. I have died to pay for all the answers to prayer that you will ever get."
Therefore, the normal way to think about coming to God in order to receive blessing is coming by the Spirit, through the Son, to the Father. That's just normal.
We should know how the Spirit works to stir us up to pray, how the Son works to purchase all the answers to prayer, and how the Father works as the great originator and giver of all things. And it's good to know that, because they did it that way together.
The Trinity worked together to make answers to prayer possible. But once you know that, that they all teamed up to do this for you, you can't not thank them! You can't not say to Jesus, "Thank you for dying for me!" You can't not sing to him.
I was just singing this morning a song (the chorus is, "No merit of my own His anger to suppress..."—that one) and it is sung about "him," and I sang it once that way. And then I just said, "I'm going to see if this works with 'you' instead of 'him.' I want to sing to you!" So I sang to Jesus this morning.
And I think the Father was pleased and the Son was pleased. And I think the Spirit was especially pleased, because he was sent into the world to magnify the Son.
When I say "inevitable," I think the saved heart so loves what the Spirit did, what the Son did, and what the Father did that he can't not not tell them.