Referring to CPU architecture, 64-bit is almost always a good thing as it allows the use of practically unlimited RAM, and even assists 32-bit applications running under a 64-bit OS; so for example, Cubase users who use large sample libraries which need to loaded into RAM will certainly benefit from the use of both a 64-bit OS and the 64-bit version of the application. Even the 32-bit version of Cubase can benefit from being run in a 64-bit OS, something which many of us are doing (usually in cases where there's no 64-bit version of a favourite VST plugin, and the VSTBridge doesn't work).
In the case of audio resolution, when audio is passed between VST plugins, it is done only after each sample is converted to what's called a "floating point" number -- think of this as mapping e.g. a 16-bit (integer) sample as it might exist in an audio file to a value between 0 and 1, but with all possible values between 0 and 1 available, thus giving infinitely more resolution during processing. Only when the result is converted back to integer samples (e.g. 16-bit, 24-bit) for playback via an audio interface or for burning to disk is the resolution decreased.
So, in summary, as far as audio processing is concerned, it does not matter whether the application is compiled for 32-bit or 64-bit CPU architecture, the audio processing would be identical because this happens in the floating-point domain.
ASIO 2.2 did not have increased audio resolution ("precision"). The article referred to  is talking about 64-bit in relation to CPU architecture, not audio resolution. The ASIO 2.2 SDK enabled developers to create versions of their existing drivers that would function properly on 64-bit operating systems. If your interface deliveres 24-bit audio with it's ASIO 2.0 driver on a 32-bit OS then it still delivers 24-bit audio with it's ASIO 2.2 driver on both 32-bit and 64-bit OSs.
Let's approach it from a different angle: if I use native OS drivers (non-ASIO) and don't load any VST effects or instruments, will I hear a difference between the same project rendered on the 32-bit version of Cubase, and rendered on the 64-bit version?
No, because the versions differ only in terms of the OS/CPU architecture they were compiled for, and the internal processing is the same.
ASIO output is "bit identical" or "bit transparent", that is, the bits sent to the sound card are identical to those of the original source
A 32-bit CPU can process a Double Precision (64-Bit) Float. This is done with integrated floating point hardware, which is often based on 64-bit units of data. For example, the x86/x87 architecture has instructions capable of loading and storing 64-bit floating-point values in memory, yet the internal data and register format is 80 bits wide.