Why do Protestants say: "In Jesus' name. Amen.", whereas Catholics say: "In the name of the Father, and of the Son, and of the Holy Spirit. Amen."?

Mostly because we like to say Jesus a lot. In the good way of course, but obviously the main reason would be that we are only connected to the Father by Jesus and therefore need to pray in His name TO the Father. The Holy Spirit is as we see it empowering us to pray in the first place and again we wouldn't pray in the name of the Holy Spirit. But also we see ourselves as connected to Jesus as the church body.

I grew up being raised Catholic, not Roman Catholic... Simply Catholic. When Catholics say "In the name of The Father, The Son, and The Holy Spirit (or "Holy Ghost"), Amen", it's a reference to the Holy Trinity. The Father IS God himself. The Son IS Jesus, who was the embodiment of God. The Holy Spirit (or Holy Ghost, depending on which you prefer to say) is seen as the spirit of God that resides within each of us. IE: All three are God, and God is all three. I hope that makes sense. I know it's a bit weird to say there is one God, yet split him up like that, but the way I understand it, it's a way of saying "God is in all things, God is all, etc". It also goes along with the "Sign of The Trinity" which is the symbol of the cross that is made with the hand as you speak.

As a side note, usually you only SAY all of this for certain prayers or particular services IN THE CHURCH. In my experience it isn't all that common for catholics to say this in personal prayers. Usually a simple "amen" will suffice. Though you will see catholics use the symbol of the Holy Trinity more often. Especially if they are the type to pray at dinner, if they observe the rosary, or observe Christmas with an Advent Wreath.

I grew up with my grandparents, who where both very devout Catholics, and my grandfather was an usher for the church, so I learned a few things. Hopefully some of what I wrote was helpful to you.

Well, as a protestant (Baptist to be specific, although I consider myself a Christian first, and the particular denomination is just a formality) we pray in Jesus' name because

1 Timothy 2:5-6 NIV For there is one God and one mediator between God and mankind, the man Christ Jesus, who gave himself as a ransom for all people.

So we pray to God the Father in the name of our Lord, Jesus Christ, because He is the reason we have a relationship with God the Father in the first place. The pastor does end his sermons with a benediction, which is a long prayer, and says "In the name of the Father, the name of the Son, and in the name of the Holy Ghost", I don't know why that is, I think I'll research it now that you mention it.

In protestant theology, it is only through Jesus' atoning sacrifice that we may enter into the presence of a holy God who is utterly incompatible with sin. We, as sinners, are allowed to pray only because Jesus died for our sins. So to say "in Jesus name" actually means "I know that I can only enter your presence because of Jesus". But of course most Protestants are pretty unaware of the theological connection and use these words out of tradition.

Although it may be that most Protestants are unaware of the actual reason and do it for silly reasons such as are cited in the other answers ("Protestants just like to say Jesus a lot, etc.") the original reason is obviously John 16:23

And in that day ye shall ask me nothing. Verily, verily, I say unto you, Whatsoever ye shall ask the Father in my name, he will give it you.

Jesus instructs that we pray to the Father in his name. To do otherwise would be like if I told you to go to a particular store and tell them David sent you. And instead you showed up at my house saying "Hi David, you sent me." Uh...no, I didn't. That's what its like praying to the Father in the name of the Father. It makes little sense. Praying in the name of the Father, Son, and Holy Ghost, is like you aren't actually praying at all, but just reciting a little speech to the congregation.

Lutherans do pray:

Pastor: In the name of the Father and of the Son and of the Holy Ghost

Congregation: Amen

during a typical service

just as it says in the online hymnal here: http://www.lutheran-hymnal.com/online/page5.html

Or see an older 1912 source:

¶ The Congregation shall rise, and the Minister, standing at the Altar, shall say:

In the Name of the Father, and of the Son, and of the Holy Ghost.

¶ The Congregation shall sing or say:

Amen.

Protestants believe that Jesus wants them to pray "in Jesus' name" and they would cite the following verses to warrant their belief:

John 14:13-14 ESV Whatever you ask in my name, this I will do, that the Father may be glorified in the Son. If you ask me anything in my name, I will do it.

John 14:26 ESV But the Helper, the Holy Spirit, whom the Father will send in my name, he will teach you all things and bring to your remembrance all that I have said to you.

It's also never stated in the bible to pray "in the name of the Father, the Son, and the Holy Spirit" which is probably why most Protestants don't pray that way--but with Protestantism you can never speak on everyone's behalf.

This passage ↓ says to baptize others in the name of the FS&HS (and that is how most Protestants baptize).

Matthew 28:18-20 ESV And Jesus came and said to them, “All authority in heaven and on earth has been given to me. Go therefore and make disciples of all nations, baptizing them in the name of the Father and of the Son and of the Holy Spirit, teaching them to observe all that I have commanded you. And behold, I am with you always, to the end of the age.”

First, it is valuable in this answer to distinguish between the pious practice of individuals, and small groups of believers, on the one hand, and the official formularies of the larger group, on the other. It may be that among Roman Catholics, there are those who use the full Trinitarian formula to begin and end an individual prayer, and perhaps every prayer they utter. However, in official Roman Catholic usage, the Trinitarian Formula cited is an invocation which begins a larger service, and not the beginning of individual prayers. Individual prayers seem most always to begin with "O God..." (in Latin, Deus) alone or with amplification. By Amplification, I refer to phrases such as "Lord God..." (Dominus Deus), "Lord God Almighty..." (Dominus Deus Omnipotens...). Individual prayers end with "Through Our Lord, Jesus Christ" (*Per Dominum nostrum Jesum Christum", again, alone or with amplification. Amplification might be (among others), "who lives and reigns with you and the Holy Spirit, World without end (in Latin per qui tecum vivit et regnat in unitate Spiritus Sancti Deus, per omnia saecula saeculorum). I have available a copy of the Latin service book of the Roman Catholic Church issued in the early 1960's, before Vatican II. I surveyed a sample of the prayers in that book, taking special care to include in the sample prayers where I would expect to see a full Trinitarian salutation or conclusion, and did not find a single example of a prayer starting or ending in that way.

Prayers in official sources (e.g., the 1662 English Book of Common Prayer, and the 1928 US Book of Common Prayer, the Lutheran Book of Worship of the mid-1970's, and the services of the Methodist Hymnal of 1933) showed the same salutations and closings on individual prayers in the services, and showed the use of the full Trinitarian formula quoted in the question only as an invocation, at least in the prayers I sampled.

I would note further, that for most protestants, ending a prayer with "In the Name of Jesus" is exactly the same as using the full "In the name of the Father, and of the Son, and of the Holy Spirit", because there is One God, comprised of the three persons of the Father, the Son, and the Holy Spirit, so that whether one uses the short form ("In the name of Our Lord Jesus Christ") or the long form, "In the name of the Father, and of the Son, and of the Holy Spirit" are both saying the same thing, as either is calling on God.