Well, I don't think anyone has studied this particular problem and the answer is unlikely to be possible on purely theoretical grounds... However, some things can be said:
- Odds are very good the the longer text will compress better [in terms of compression ratio, not in terms of absolute length, of course] no matter if it is adult or child literature. (A point already mentioned by Peter Shor in a comment above.)
- Even the best LZ-based text compression algorithms (which use actual words unlike the run-of-the-mill zip) get beaten by stock bzip2 on text compression ratio; bzip2 does something seemingly "very dumb" [actually very smart] in that it [reversibly] scrambles the text into gibberish stings but with many repeated letters; this is the Burrows-Wheeler transform. I do not expect that the frequency of words will matter much for bzip2-style compression, but instead the letter frequency probably would; it is much harder to guesstimate if there's a difference with respect to the latter between child and adult literature, but I suspect they don't differ substantially on letter frequencies.
Both points can be seen [on general texts, not child vs. adult] in tables 2-3 (p. 43) of the paper "Text Compression: Syllables"
by Jan Lansky and Michal Zemlicka (sorry for leaving out the diacritics in their names). Mind you, this is for English and Czech text [beware that table 3 has an incorrect caption, "bytes" should be "bits", same as in table 2], and you can see that bzip2 does less well on the latter almost surely because there are more symbols in base alphabet for Czech compared to English. So for (say) Chinese (using a native encoding, not via Romanization), I would not bet on bzip2 anymore...
Strangely enough, Salomon's Handbook of Data Compression [5th ed. 2010], where I found ref to the aforementioned paper and a fairly ample [re]presentation on pp. 1122-1127, doesn't mention the sequel [which was promised in the conclusion section of Lansky and Zemlicka]: Syllable-Based Burrows-Wheeler Transform (2007). Alas the results are underwhelming: you need fairly large documents (>200-500kB) for syllable-based BWT to overtake the letter-based BWT, and even the then the improvements are fairly small. The word-based BWT, also tested in there, never wins, although the max document size was 5MB. I think this shows that word repetition is not the major factor in the text compression ratio (if done right) compared to letter and syllable repetition (frequencies), at least for the usual book-sized documents. So I think answers the question negatively (as in probably no difference) for same-size documents and only some difference in word repetition (like child vs. adult literature would entail) when using stock bzip2 or the experimental syllable-based BTW. Perhaps if you compress something much more vast like the whole Wikipedia (or to particularize to this case, two vast collections of literature, one for adults for one children), then word repetition might be the key factor in getting good compression (and you would want a word-based BTW).
A couple more twists to this:
The research cited above assumes a single choice of syllable/word/letter as "predictor". Context mixing (CM) algorithms (pioneered by PAQ) actually use several (adaptively weighted) predictors; for example PAQ use both a previous-word and bit-level PPM predictor (among others).
The previous word may seem rather unlikely to work much, but on a 1GB sample of Wikipedia, it actually works quite well because there are some bot-generated articles with templated text, like those based on US census data. (See the 3rd image, "repeated string analysis" and commentary in http://mattmahoney.net/dc/dce.html#Section_22. Finding all of them requires a large amounts or RAM though; the best performing derivative of PAQ [called cmix] on that dataset uses ~30GB of RAM.) I would expect a CM algorithm [which uses previous words as one of the predictors] to do better on texts with smaller vocabularies (like children stories) all other things being equal.
If you're willing to apply text-only transforms (BWT is not so), then you can do better than BWT; see the (somewhat improperly named) "word replacing transformation" (WRT) paper which actually consists of a whole bunch of little tweaks that make words matching each other more likely, like case conversion, replacing [in a reversible manner] end-of-lines with spaces, inserting spaces before punctuation etc. It doesn't seem like any of that would favor smaller vocabularies, but there are a lot of techniques listed in there and I haven't read that paper thoroughly.
In a way combining the two previous points, one can actually use a word-based (word as in natural-language word) compression as a pre-processing step for a general-purpose (including LZ-based) compressor. I'm a bit surprised this hasn't been tested much before the 2009 paper by Farina et al. This idea does pretty well when the base compressor is gzip, but the compression ratio improvements for bzip2, 7zip, or PPMd as backends are rather modest. Perhaps the most notable thing about this precompression is that it increases overall compression speed. Percentage-wise, this was most notable for 7zip as it saw a 3x-4x compression speed improvement! 7zip is LZMA-based, so roughly speaking LZ on steroids: a very large dictionary and with range encoding instead of Huffman used in gzip/DEFLATE. (Tables 6-8 have the gist of the results.)