How were Jewish sacrifice different? When reading the Old Testament there are many references to sacrifice being required as atonement for sin. I find that today, we commonly nod along to all the detailed instructions and remind ourselves that Jesus made it so we don't need to do that anymore.

I can't help but feel like we are seriously missing some context. In many of the things Christians (and Jewish people before us) do, we are being set apart from those around us.

In what way were the sacrifices being offered in the Old Testament "set apart" from what was going on in other contemporary religions and cultural practices?

For example:


Did others not remove the organs? Thus, Jewish people doing this would be an outward sign that they were not sacrificing to the local gods?


Were the organs and fat prized as food for others?

Was the method of sacrifice significantly different than preparing food, so that fellow Jewish people would clearly know the animal was for sacrifice, and not regular food?
Was the value of what was sacrificed by others significantly more or less?
I will answer as much of your questions as I can, some of them need be asked on other sites. For instance I do not know what ritualistic sacrifices were in other religions at the time, and most Historic questions will be better answered on Biblical Hermeneutics.
Some things to keep in mind are:
These sacrifices; given to the Hebrew nation were given prior to Jesus time on Earth. Therefore, there had been no sacrifice of human life which is indicated in the following Scripture:

Genesis 2:17 KJV But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.

It is significant that this command was only given to one person that being Adam. Man was the only creature that was made in the image of God himself, and his life was therefore, more significant than God's other animalistic creations. 

Genesis 1:27 KJV So God created man in his own image, in the image of God created he him; male and female created he them.

The reason this is significant lies in the fact that Man had become a living Soul.
The Soul is an eternal being, So man has an eternal part, and yet God said that if Man disobeyed him he would have to taste death. 

Genesis 2:17 KJV But of the tree of the knowledge of good and evil, thou shalt not eat of it: for in the day that thou eatest thereof thou shalt surely die.

The Hebrews had just left Egypt, where they had served other gods, and although it is spelled out, it seems that they would also make unholy sacrifices to those gods.
They had also taken many riches from the Egyptians, God did ask for the sacrifice of those riches. What he did ask for was a sacrifice of life, and it must be remembered that God had saved the cattle of the Hebrews, when he killed Egyptians cattle along with the first born of the Egyptians.

Exodus 12:12 and 13 KJV For I will pass through the land of Egypt this night, and will smite all the firstborn in the land of Egypt, both man and beast; and against all the gods of Egypt I will execute judgment: I am the LORD. 12:13  And the blood shall be to you for a token upon the houses where ye are: and when I see the blood, I will pass over you, and the plague shall not be upon you to destroy you, when I smite the land of Egypt.

Cattle were at the time a measure of wealth. So God's sacrifice was not only a sacrifice of wealth but also a sacrifice of life, as was demanded for disobeying God's commands.
The sacrifice ritual was so different from food preparation, in that it required actions not normal in the normal killing of livestock, and only certain parts of animals, not sacrificed for a sin offering could be eaten, and then only by a chosen few.

Exodus 29:10 thru 14 KJV  And thou shalt cause a bullock to be brought before the tabernacle of the congregation: and Aaron and his sons shall put their hands upon the head of the bullock. 11  And thou shalt kill the bullock before the LORD, by the door of the tabernacle of the congregation. 12  And thou shalt take of the blood of the bullock, and put it upon the horns of the altar with thy finger, and pour all the blood beside the bottom of the altar. 13  And thou shalt take all the fat that covereth the inwards, and the caul that is above the liver, and the two kidneys, and the fat that is upon them, and burn them upon the altar. 14  But the flesh of the bullock, and his skin, and his dung, shalt thou burn with fire without the camp: it is a sin offering.
 Exodus 29:27 and 28 KJV  And thou shalt sanctify the breast of the wave offering, and the shoulder of the heave offering, which is waved, and which is heaved up, of the ram of the consecration, even of that which is for Aaron, and of that which is for his sons: 
Exo 29:28  And it shall be Aaron's and his sons' by a statute for ever from the children of Israel: for it is an heave offering: and it shall be an heave offering from the children of Israel of the sacrifice of their peace offerings, even their heave offering unto the LORD.

These sacrifices were a precursor to the one true sacrifice, which is Jesus. Even though man has an eternal soul that Soul will not necessarily spend its eternity with God in Heaven. This is because life is necessarily sacrificed to God in exchange for our disobedience. That requirement is not possible because we have inherited the knowledge of good and evil, and unfortunately, we appear to have only enough knowledge to more often ignore God's requirements in favor of our own desires.
Christianity is the knowledge that:

John 3:16 and 17 KJV  For God so loved the world, that he gave his only begotten Son, that whosoever believeth in him should not perish, but have everlasting life. 17  For God sent not his Son into the world to condemn the world; but that the world through him might be saved.

All sacrifices both in the Old Testament and the New Testament serve to bring the world back into the good graces of God.
Hope this helps.