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.
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:
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:
And Peter says that every believer is called to a holy priesthood, in which we make all manner of sacrifices to God
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: