Are sacrifice and prayer the same?
The word sacrifice can mean different things in the Bible depending on the context. In the context of the Law of Moses, a sacrifice refers to the killing of an animal as a blood atonement for sin offered by the priests.
Ex. 20:24 ‘You shall make an altar of earth for Me, and you shall sacrifice on it your burnt offerings and your peace offerings, your sheep and your oxen; in every place where I cause My name to be remembered, I will come to you and bless you.
In both the Old Testament and the New Testament, the word sacrifice can also mean any act of devotion to God, as when the Psalmist writes:
Psa. 27:6 And now my head will be lifted up above my enemies around me,
And I will offer in His tent sacrifices with shouts of joy;
I will sing, yes, I will sing praises to the LORD.
The psalmist compares singing praises to God with making a "sacrifice" of joy. In the New Testament, Paul compares our financial gifts to ministry with a sacrifice:
Phil. 4:17 Not that I seek the gift itself, but I seek for the profit which increases to your account.
Phil. 4:18 But I have received everything in full and have an abundance; I am amply supplied, having received from Epaphroditus what you have sent, a fragrant aroma, an acceptable sacrifice, well-pleasing to God.
And Peter says that every believer is called to a holy priesthood, in which we make all manner of sacrifices to God
1Pet. 2:4 And coming to Him as to a living stone which has been rejected by men, but is choice and precious in the sight of God,
1Pet. 2:5 you also, as living stones, are being built up as a spiritual house for a holy priesthood, to offer up spiritual sacrifices acceptable to God through Jesus Christ.
So in that sense, prayer can be an act of sacrifice to God, in that it is an act of devotion to God, but it's important to distinguish these forms of sacrifice from the original form under the Law of Moses. The sacrifices under the Law were mandated as a means of covering sin (temporarily), but now that Christ has died for the sins of the world, there is no longer the need to perform literal sacrifices (i.e., killing animals).
Therefore the sacrifices we perform now (i.e., our praise, giving, prayer, etc.) do not obtain our forgiveness from God. They are made out of a love for God and a desire to obey Him and please Him. Paul sums up our obligation this way: