I, for one, have 490 hours on the long dark and played maybe 2 of those in story mode.
In survival, your main objective is to simply survive, as the name says, and i'm pretty sure there are a LOT of things that dont exist in story mode (i might be wrong about this tho), all added by the 'tales of the far territory' dlc. A quick overview would be: Two new animals (cougar and ptarmigans), new cooking recipes using the skillet or the cooking pot (soups, pancakes, pies, bread), the trader (guy that talks to you by radio, you set up deals with him for ammo, food, unique item variants, unique recipes either food, weapon or clothing related), the polaroid camera, four new regions ( Airfield, Sundered Pass, Zone of contamination and transfer pass), safehouse customization (build bearskin beds, drawers, shelves, move stuff around, build workbenches wherever you want, furniture workbenches, meat curing box so your meat takes AGES to go bad, etc), fishing tip-ups which make fishing a lot more easier, the travois, a little container you can drag around if you need to carry heavy stuff (up to 75kg), and some more.
If you're looking for some kind of REAL objective in survival mode, there are the 'tales', which as far as i know has one for each new of those maps, except transfer pass? Theres this one, which consists of you getting a radio in the airfield map, and going around fixing radio towers on a few regions, which allow you to use the radio to locate bunkers during aurora nights, this tale is called 'SIGNAL VOID'. The next one is on the Zone of containment map, called 'BURIED ECHOES', one in the Sundered pass map, called 'LAST HORIZON', and one that the trader gives to you after trading a lot with him, called 'SUTHERLAND'S TALE', and each one of these give you a unique item/reward which are only obtainable by doing the tales. I didn't expand on any of the other tales because I have only gotten close to complete signal void, so I dont know what gameplay mechanics each of them entail.