Vulkan is a cross-platform 3D graphics and computing API that is meant for high-performance 3D graphics applications focused on modern hardware. DirectX is also an API, but it is focused on Windows.
One can say that DirectX allows the software to better communicate with the hardware, allowing developers to better optimize their games. Regarding which is better is not a new debate, and it is also nothing new in the world of PC gaming. The debate about Vulkan versus DirectX predates Valheim. Either way, there are good and bad reasons for players to use either of them when playing Valheim.
6 DirectX Is Safer
If in doubt, just go with DirectX. Chances are, it will work just fine. Yes, Vulkan is an API meant to work well with modern computers and hardware, but DirectX was designed to work. If players use Windows and are not too sure of which one to go for when exploring the most dangerous biomes in the game, DirectX is most likely the best choice for them.
The most recent DirectX is trying to catch up with Vulkan, so chances are, differences will not be so easy to notice. Although there are reports of performance issues on DirectX12, it seems that, so far, Valheim is only using DirectX11. It means that they are playing safe, which is nice since Valheim is not the most demanding of the games out there. Still, Microsoft products don't generally remain unpatched for too long anyway.
5 Vulkan Is Meant For Modern Hardware
The APIs that we have in the market have been evolving from their past selves and catching up with the new hardware and trends just fine. But the interesting thing is the fact that they're catching up. Vulkan's intent was to be ahead. It was already designed for modern hardware and therefore has a greater chance to work better with modern 3D games.
Although keep in mind that there is always a chance of incompatibility and unseen interactions that might make the game run a bit slower on a specific setup. Trying it out does not hurt if, so it is recommended for those who can.
4 DirectX Is A Better Choice If You Are GPU Restrained
In case a player does not have the best GPU to play Valheim, but the CPU does the job well, the recommendation is to go with DirectX. Generally, not having the newest GPUs will be a greater challenge for Vulkan anyways. It might be a matter of how the game was coded and optimized. Either way, DirectX does make more effective use of weaker GPUs than Vulkan is capable of doing.
Vulkan makes a setup's GPU do most of the heavy lifting, letting the CPU working less. If players are asking themselves which to use, but have an awesome or modern GPU, it might be better to just go with Vulkan.
3 Vulkan Is The Obvious Choice For Linux
Linux users always have to go through a little more trouble in order to play the games they want. Generally speaking, the Windows emulator with DirectX was the solution for many years. Now thanks to Project Mesa, also called The Mesa 3D Graphics Library, a software implementation of OpenGL, Vulkan, and other graphics APIs, Linux users have an easier time.
So if players have windows, sticking with DirectX is taking the safer, simpler approach. But if they use Linux, then most likely the choice that will make things easier is going with Vulkan. That way, there are fewer worries about compatibility issues, windows emulating, and other things that users of systems other than Windows usually have to worry about.
2 DirectX Is A Comfortable, Safer Choice
The better choice here will be entirely dependent on the system. One must consider compatibility issues that a setup and system might have with certain APIs, for instance. Maybe Vulkan makes a computer heat up slightly more than DirectX, causing performance issues after a while of gameplay. Or maybe, one specific update of the Graphics Card made DirectX not to work properly for a couple of months. Things are not always perfectly clear when it comes to PC gaming, and Valheim will not be an exception there.
The general rule remains the same, though. Vulkan is more likely to put most of the weight on the graphics card, while DirectX, for now, is more demanding on the GPU instead. Unlike the origin of the Ulf, going with DirectX is known to be a safe choice.
1 Vulkan Can Surprise, But The Difference Is Not Huge
Most videos of people testing the game with Vulkan and DirectX will show very similar results. There is, of course, a slight difference in FPS, and hardware usage, but the performance, in the end, tends to be too close. It can be seen as an unpopular opinion about Valheim, but there is not enough difference to declare one as considerably better than the other.
This is one of the reasons why experimentation is recommended when possible. Not only that, but it is also a good reason not to worry too much about it if the game is already running well. Most players will barely perceive any difference. Unless there is a special interest in improving the performance by a couple of frames per second, either launcher should most likely be okay to play the game.