Here's a late response, but this question keeps taunting me for a direct and difficult-to-craft answer.
Is there a significant difference in the rigor of these courses as
taught in college and high school (i.e., is it not just my
Likely yes; although this will vary by location, institution, and instructor.
If there is a difference, is there any published research supporting
such a difference, or statements by those in charge of curriculum at
an institution giving an explanation?
My best reply is this: The published research supporting this difference is the entire body of literature for the discipline of mathematics since Ancient Greece.
To expand: College professors are likely to be active mathematicians and have a perspective on the structure and development of mathematics that high school instructors are less likely to have. Moreover, college professors are principally incentivized and promoted on the basis of doing math, whereas modern high school teachers are judged on the basis of passing students. At the high school level, the algebra course is likely terminal for most students; whereas at the college level, it will be viewed as only the beginning of a much larger body of knowledge and exploration. The college faculty will want this foundation to be firmly set, with correct explanations and formulations, so as to support later work throughout the discipline. At high school this is unlikely to be a goal or even an observation; if the system can get the student past a given standardized exam, it is considered "job done".
"But perhaps the greatest accomplishment during the first three
hundred years of Greek mathematics was the Greek notion of a logical
discourse as a sequence of statements obtained by deductive reasoning
from an accepted set of initial statements assumed at the onset of the
discourse... Certainly the most outstanding contribution of the early
Greeks to mathematics was the formulation of the pattern of material
axiomatics and the insistence that mathematics be systematized
according to this pattern. The concept of axiomatic development in
mathematics must be ranked as one of the very greatest of the GREAT
MOMENTS IN MATHEMATICS." -- Howard Eves, Great Moments in
Mathematics (Before 1650), Lecture 7.
To the college professor, math without a logical axiomatic development doesn't even count as mathematics at all. Personally, I refer to the memorize-this-formula-without-proof approach as "faith-based mathematics".
Now, if you look for some modern literature on teaching algebra at the college level, it will be very easy to find publications arguing exactly the opposite. That is: Most students won't "use" algebra, practical applications are the only thing we need to consider, deep understanding is extraneous, taken-for-faith procedures are acceptable, college professors should be removed from remediation efforts, etc. But this overlooks the whole point of mathematics -- the search and sharing of proper justifications of our knowledge. This oversight is exactly how rigorous math has been removed from high schools in the last few decades, resulting in the jarring wake-up call when students jump the chasm to college-level work. There is definitely a movement to remove real mathematics from college academics for the general student in the future, because of this ongoing difficulty; but the idea that most students will never even be introduced to real mathematics is almost overwhelmingly difficult for the dedicated math practitioner to swallow.
In short: Math without logical proof is not really mathematics at all. The fact that today's high school programs leave the majority of students completely unaware of that fact is indeed an ongoing tragedy, one that we can only hope won't be forced as a contamination into our higher education programs.