I think that rather than a difference between dialects of English, there is a difference in meaning. Small and little are not always synonymous (see this thread and this BBC page for more general discussions). Generally speaking, small tends to be more literally about size while little can be more metaphoric.
In the case of a child, small refers directly to the child's size. A small child is one whose size or height is small. A little child more directly evokes the child's youth. The two can be synonymous, because age is strongly correlated with size, but they carry different connotations. I would expect a child to say “I'm only little”, implying primarily that they are young, rather than “I'm only small”. On the other hand, a child might say “I'm small, I can squeeze through”, perhaps more than “I'm little, I can squeeze through” (though the sentence with little is also idiomatic).
Comparing “only a little child” with “only a small child” in British English and American English doesn't reveal a significant difference between the sides of the Atlantic. It does show evolution over time: small was practically unheard of in the 19th century and is now on par with little.
A related adjective is short. It relates directly to the person's height, and tends to imply short for their age.
Jimmy Kimmel Live is not a live, spontaneous television program Wikipedia. So, my first thought was "better ask the writers of the program".
While television is usually not "real", a second thought was "there might be something here".
The comments on this question are good, and I determined to ask some questions.
There is some possibility that many Americans do not readily remember exactly how they were evaluated in school.
Older people I asked first responded "tests" when asked about evaluation in school. Younger people the same.
But further questions had the older people remembering "quiz" and "exam" also. Younger people tended to stay on "test".
Other inquiries produced an anecdote about a 15 year old girl who laughed when her father said that a "limb" had fallen on a walking path. She was sure a "limb" was an arm or leg.
There is nothing definitive in what I learned. But it may well be that English as used by Americans is evolving faster than some of us might think. It may be getting simpler. That may explain how the writer or writers of Jimmy Kimmel Live came to this attempt at humor. Maybe "exams" are not such a conscious part of some American's schooling today. Or, this whole business might be some fluke.
The answer to this question has to come from those who originated the words that motivated the question. My first thought seems validated. But, as there is little hope of getting the required information from the actual source, the continuing thoughts I had may help explain things a little.
The end answer has to involve those who created this "humor" I am sure there is no recourse to finding a definitive answer beyond the ultimate motivation for the question, those who are responsible for the program.
Best Answer
The British authority on such matters is Debrett's. Recommendations on the wording for wedding invitations are given here.