What's the difference between a simple and 1-foliolate (unifoliolate) leaf?
Short answer:

Look for a pulvinus at the leaf-stalk/leaf-lamina junction.





Long Answer:

So from a strictly anatomical identification standpoint, this can be a bit confusing. 

According to "Plant Identification Terminology: An Illustrated Glossary" (Harris &amp; Harris, 2001):


  Simple: Undivided, as a leaf blade which is not separated into leaflets.


Where:


  Leaflet: A division of a compound leaf


and


  Compound Leaf: A leaf separated  into two or more distinct leaflets. 


So the definition of a compound leaf is to have "two or more distinct" leaflets. 

But wait...

From plantnet:


  one-foliolate (1-foliolate, unifoliolate): a compound leaf reduced to a single leaflet, usually recognized by the articulated or jointed ‘petiole’, which is in fact a petiole plus a petiolule.


So what gives?

Perhaps this is due to a deficiency in terminology, as discussed (and corrected) by Jiménez-Saa 2011.

But, again Harris &amp; Harris (2001) hint toward how this might arise [emphasis mine]:


  Unifoliolate: a leaf theoretically compound, though only expressing as a single leaflet and appearing simple.  


So there is still a fundamental difference between the two. 

Structurally, we can see the difference at the basal joint of the foliar structure:


  Simple or unifoliolate: The leaf is undivided...with (unifoliolate) or without (simple) an articulation between the leaf lamina and leaf stalk. [source]


Put differently:


  Unifoliolate leaf is usually recognized from a simple leaf by the presence of a distinct joint or a pulvinus at the lamina-petiole junction, which is in fact a petiole plus a petiolule.



Note: Pulvinus = a distinct enlargement/swelling at the base of (sometimes at the apex of) a petiole or of a petiolule. 




Note: a discussion about the developmental evolution of unifoliolate characteristics in Cercis canadensis is discussed here.