Все вещи, которые нас не убивают, делают нас сильнее
Все вещи, что нас не убивают, делают нас сильнее

Both are possible. But as for me the difference between который and что there is really similar to the differences between English which and that.

For example:

My car, which is red, is fast. --- Here the which clause means that the information there is secondary. Like everyone knows that you have only one car and you just add that it's red.
My car that is red is fast --- Here the that clause adds crucial information. It's like people knows you have more than one car and you differentiate between your cars by using "that" and it's like you pinpointing your particular red car with "that"

Although, Russian language doesn't really provide the exact same difference between что and который there but it's still very similar. Examine these:

Все вещи, что нас не убивают, делают нас сильнее - This что doesn't really target the respective noun вещи, well of course it does but not that strong. It's like что is only used to get clauses connected properly. You don't empesize the respective noun in any way. And these вещи seem to be just all the things in the world, not particular ones.

Все вещи, которые нас не убивают, делают нас сильнее. This которые does target the respective noun and does so very strongly. It does do exactly the same thing is "that" in that English sentence. You're really pinpointing the respective noun вещи with которые there. It almost sounds like these вещи are a particular set of things, not all the things in the world like in case with что.

Mind you that I did exaggerate the difference between что and который in there but sometimes a good well-placed exaggeration is a great tool to use.
So I do hope it helps in any way

Good luck with that