I think that's just it -- lots to wrap your head around.  this one video was a real revelation to me.  
I just searched and found out the 22 has no rear inputs - that's the 44.  my bad. 
the ur22 does require an unbalanced cable on input 2 for direct instrument recording.

noynekker
Thanks swamptooth, an interesting video . . .
- no distortion in the DAW because it uses floating point maths
- the signal only distorted when converted to integer format to send to the audio interface
- not possible to clip a modern DAW's mix bus internally . . . consider it to have infinite headroom.
 
Lots to think about here, seems I really need to pay closer attention to my soundcard mixer's I/O settings.
 
As to the OP's original question, is it possible that the quality of the newer soundcards analog - - -> digital conversion is of a lower grade, causing the more noticeable distortion. What else could it be, unless the I/O settings have been changed in the new set-up ?

But it IS possible to clip any plugins you've got if the input goes over a certain threshold.
This threshold will vary from plug to plug, so keep an eye on whatever metering provided by the plug(s) you're using.

Its a very good idea to keep all tracks and buses out of clipping. As Bristol Jonesey points out not all plugins can handle a hot signal. 

You also want to make sure that your final mixdown isn't too hot. The issue that the people listening to your works hear may be due to clipping in the D/A or preamp of thier playback device caused by not leaving enough headroom in the final mix.

To answer the original question, yes - different interfaces can sound different when clipped.
 
Some interfaces are more prone to analog clipping than others, so the same signal level to one might cause more or less clipping than another. Furthermore, some interfaces are well-protected against analog clipping but end up passing the overly-hot signal on to the ADC and get digital foldback instead, which is much more noticeable than analog clipping.
 
You just have to learn your particular interface, discover its limitations and work within them.
 
None of this has anything to do with clipping within the DAW. If you experience distortion in a plugin, your audio interface won't make it better or worse. Beware final outputs exceeding 0dB, though, as that will translate into digital overs when converted to integer data, and that really does sound bad regardless of the interface.

bitflipper

All else is in doubt, so this is the truth I cling to. 
 

I've asked before; but, what is the significance of this "Sin-X-over-X" expression?  

Any physical waveform may be represented over the interval -∞ < t < ∞ by that equation. It has been known to mathematicians since 1915 and was brought to engineer's attention in 1949 by Shannon. Also known as the cardinal series. It is the basis of the Sampling Theorem.

Well ... thank you!  I've long ago forgotten what I learned about Shannon, Nyquist  and sampling. Thanks again.

mettelus
Do the wave forms of the tracks you are referring to have flat tops and bottoms? This is a key indicator of clipping regardless of what the interface is telling you.

Maybe what I am hearing is not "clipping".  The waveforms are xxx flattened as one would expect with clipping.  Here's a link to a sample that has the "noise" I hear.  The sample is marked in two places BUZZ1 and BUZZ2.
 

I'd take a look but I don't have the plugs that you're using.  is it happening at the project level or could you post just an audio track that has the noise??