Both are bit-perfect.
ASIO vs WASAPI (WDM) is a question with no general answer. Often it makes no difference at all. Sometimes one does better than the other, usually because the driver author did a better job on that one.
The link @philr posted was good. I also wrote about this recently here…copy-pasted and cleaed up:
I think Microsoft did a good job with WASAPI and that alternative driver stacks like ASIO are for the most part an unnecessary evil in 2015.
There are some wrinkles, however…
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Native DSD–when a DAC supports it, ASIO drivers can enable direct transmission of DSD data without encapsulation. Both options are bit-perfect, but some people swear that “Native” transmission sounds better.
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Some USB interfaces expose a non-bit-perfect volume control via their WDM drivers. You can tell you have one of these if DSD sounds like a hiss at <100% volume. If you have one of these, disable Roon’s volume control (which is just tweaking the WDM volume controls under the hood). ASIO avoids this problem by having no support for volume control at all…so if you use ASIO you lose the ability to control volume within Roon and must use the controls on your dac/amp.
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Some manufacturers do a better job on one driver or the other. Sometimes the drivers talk to the card very differently and can expose different hardware behavior. Fragmentation is sloppy.
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Manufacturers are chronically terrible at exposing accurate descriptive strings for their devices, and often make weirdly inconsistent choices between their ASIO and WDM drivers, and this can make the settings/configuration experience more confusing than it needs to be.
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ASIO is an option to try to solve a problem with a bad WDM driver, but I always start with the non-ASIO option first.