Re: Is there a difference between 3'untranslated and 3'noncoding gene region.

No. Both terms refer to the region found in a mRNA downstream of the 
stop codon for the gene, sometimes also referred to as the "trailer"
region.

Sometimes you will see 3'noncoding region to refer to the untranscribed
region after a gene, but this is more appropriately termed as untranscribed
spacer region to distinguish between the transcribed but not translated 
region vs the not transcribed region.

I hope this helps.

Michael Benedik
University of Houston