Which stats/perks are more important in new survival mode?

Many other perks have become much more useful with the changes in the new survival mode, especially in the endurance tree. As the mode name implies, survivability is much more difficult, and can lead to loss of progress due to not being able to save at will. On top of the ones you've listed (Strong Back and Local Leader):

1. Toughness, Refractor, and Lifegiver: On top of Endurance giving you more overall health, many of its perks increase your survivability in the Commonwealth. Since saving is pretty much strictly tied to using a bed, loss of progress is a real threat, especially due to the big increase in damage taken. These might give you that little extra needed to survive. Toughness and Refractor increase damage resistances, Lifegiver for +60 health and slow health regeneration (at rank 3).
2. Adamantium Skeleton: Limb damage seems to be more of an issue. Limbs now require a Stimpak to be healed, even outside of battle. Being over-encumbered can also cripple your legs.
3. Lead Belly: Since you need to keep hydrated, it might not be a bad idea to pick up Lead Belly. Keeping rads as low as possible is more urgent now, as RadAways now add fatigue and an increased risk of illness by using them, so they need to be used more calculatedly. Which leads us to...
4. Rad Resistant: The less rads you take, the better, due to the negative effects of RadAways. Depending on which DLCs you have installed, you may have a better way to manage radiation long-term (such as the Decontamination Arch offered in the Wasteland Workshop DLC), but resisting extra radiation in the field can still be useful, especially if you're spending some time in the Glowing Sea or fighting Children of Atom with Gamma Guns.
5. Solar Powered: Even though this is at 10 endurance, it can be simply great for maintaining health. The second point reduces rads in the sunlight, the third recovers health. It can save you from using some Stimpaks, and prevent you from needing RadAways in many cases. You don't have to worry about the ill effects of RadAways if you don't have to use them. This can also assist in healing quicker to counteract the slowed recovery from food/Stimpaks.
6. Cannibal: You can eat humans, ghouls, or super mutants for an immediate, decent health boost. Save Stimpaks/food, no need to wait for the slow recovery. Cannibalizing a fallen enemy will also recover some of your hunger. However, this seems to come at a steep cost, according to this answer. Cannibalizing an enemy will give you the Dark Craving debuff, which allows you to only eat human flesh (no standard food) to recover health and hunger. Moreover, it seems the rate at which your hunger rises is greatly increased, forcing you to seek out more cannibal victims until the debuff is removed (by sleeping a long time, 24 hours seems to do the trick). Depending on how you play, this might not be worth it.
7. Party Boy/Party Girl, Chemist, Chem Resistant: If you aren't adverse to using alcohol/chems, these would be useful. They increase the duration of alcohol/chems, give you immunity to addictions, and can raise your luck (Party Boy/Girl). The immunities are especially noteworthy. Illnesses are more likely to happen to those in poor health. If those ill effects are stacked with withdrawal effects, you might be severely hampered.
8. Medic: Making your Stimpaks/RadAways heal more is never a bad thing. The last point also makes them work faster, which will counteract at least a bit of the "slower healing" from them. Healing more can also help conserve Stimpaks, which are needed more overall. As previously mentioned, they (or sleeping) are needed to heal limbs. Followers that are downed also need Stimpaks to recover, else they will go back home.
9. Lone Wanderer/Inspirational/Attack Dog: Depending on how you want to play. Lone Wanderer decreases your damage taken, and increases your damage dealt and carry capacity, when without a companion. Inspirational does all the same for your companion, increasing their survivability. Attack Dog can have Dogmeat hold down enemies, preventing them from attacking you.
10. Animal Friend/Intimidation: Again, if you wanted to use this playstyle. Why fight enemies when you can avoid confrontation/make them fight for you?
11. Nerd Rage!: Damage resistance bonus and extra damage inflicted at low health. The damage resistance may save you in a pinch.
12. Action Boy/Action Girl, Grim Reaper's Sprint: Especially if you need to sprint often or are using VATS more often. Fatigue affects your AP bar the same way radiation does to your health bar. These help ensure you have some AP to use when you really need it.

Many other non-S.P.E.C.I.A.L. perks can also be handy. Astoundingly Awesome Tales, Wasteland Survival Guides, and Unstoppables all have great perks that reduce (or ignore) incoming damage, bonus healing from foods, regenerating health, etc. Live & Love increases the usefulness and survivability of your companions, should you choose to use one.

There are probably several more past these that can help you in a run through the new survival mode, depending on how you play. The ones above can really help you survive and prevent the dreaded loss of progress from death.

Sources:

Fallout 4 Perk List

Bethesda: Fallout 4, changes to Survival Mode

Around level 19 now. Here are a few I've found helpful:

Lone Wanderer - invaluable, especially with the carry capacity boost.

Ninja - I'm sticking with a sniper/stealth build, so it's helpful to cause max damage from as far away as possible.

Gunslinger - due to many factors, it's harder to gather junk, and since pistols are easier to upgrade at lower ranks for Gun Nut, I started with Gunslinger instead of Rifleman. Will add that later.

Local Leader - you'll want to get this going as soon as you can. Once you are out in the wild, it's very helpful to have access to all of your workstations, especially with no fast travel and limited carry capacity.

Demolition - explosives have been very important for me thus far, and making them more powerful and having the ability to craft them, is even better.

Oddly enough, I don't think I've taken any of the first 8 perks listed above. Started with STR 1 and END 2. Just a different playing style.

I have a lot of playtime on Fallout 4. This includes beating the original survival difficulty in less than 6 hours of game time, below level 20.

I'm now nearly level 30 in the new survival mode, taking my time. I'm on my way to the Glowing Sea.

So far, I haven't had to change my play style barely at all. The combat is essentially the same. The best way to survive is still to kill everything in VATS before it gets to attack you. The extra damage actually makes this easier. But, it also makes the whole killing them before they attack that much more important. So, continue to focus on doing spike damage with crits, and using Awareness to ensure that the damage from each shot matches the target.

This means that the most important SPECIAL stats continue to be Perception (accuracy), Agility (AP), and Luck (crits). My character has 2 Strength and Endurance, and no perks from either of those columns.

The only real differences I've found that affect my perks are the need to carry around additional supplies in the form of food and water, and the inability to fast travel back to my main settlement. This means that you need more carry weight, and more access to crafting supplies and stations. Therefore, Local Leader is the perk that has had its importance increase the most.

You need supply lines to your forward settlements so you can

• sleep (save)
• heal your radiation (http://fallout.wikia.com/wiki/Decontamination_arch)
• cook food
• drop off junk without running all the way back to your main settlement

Also, Local Leader is required to build stores in your settlements. This means you won't have to run for ten minutes just to sell your loot.

Due to the carry weight issues, you might consider Strong Back or Lone Wanderer, but there are other ways to address this issue beside wasting perks on it. You can use pocketed or lightweight gear, be more judicious about what you pick up, and take advantage of a companion. See this other answer for more info on increasing carry weight.