After some research, there seems to be little difference in meaning. In some situations, maybe ease of pronunciation is more of a guide than nuance of meaning. I was informed that 大きな木 would be preferred to 大きい木 and a quick Google search shows that these key phrases yield 2,990,000 and 119,000 results, respectively. (But 木 is clearly no abstract concept.)
However, my question stated that 大きい etc. can be used as both イ- and ナ-adjective, which seems to be only half of the truth. It depends in what sense you use the word ナ-adjective, but if you take it to mean the concept 形容動詞 from Japanese grammar, the word 大き does not qualify as ナ-adjective.
The reason for this is that 大き cannot be inflected, to form an adverb, for example. (Cf. 静か -> 静かに.) For this reason, 大きな belongs to the class of [連体詞]{れんたいし} (Rikaichan calls this pre-noun adjectival or adnominal adjective).
連体詞 are words which cannot inflect and must modify a noun (or pronoun), e.g.
この、その、あの、……
おかしな、大きな、小さな、……
as in
このやり方は一番よい。
こうするのが一番よい。
or
空におかしな形の雲が浮かんでいる。
空に変な形の雲が浮かんでいる。
These 連体詞 seem to come in 4-5 varieties (depending on the book you read):
「~の」の形・・・この、その、あの、どの、ほんの
「~な」の形・・・大きな、小さな、おかしな、いろんな
「~た、~だ」の形・・・たいした、たった、とんだ、ばかげた
「~る」の形・・・ある、あらゆる、いわゆる、さる、きたる、いかなる、堂々たる
「~が」の形・・・わが
All this information is a summary of this site. I hope someone will find it useful.
This should also shed some grammatical light on this and this question.
In an after-thought as to why 小さい時 cannot be 小さな時 is that 小さな is a 連体詞 and must modify a (pro)noun and thus forces the meaning small time, rather than allowing for the interpretation as a sentence in which the (omitted) subject was small, which is usually translated with a relative clause, i.e. the time, when I was small or just when I was small.
So, the general rule seems to be that one can choose either one of the word pair, but when choosing the one ending in な, one chooses to modify directly the noun which must follow it and disallows constructions with relative clauses.
E.g. compare
山の大きい街 vs. 山の大きな街.
The former can mean a city with a big mountain, whereas the latter is necessarily a big city situated on a mountain. (One can probably come up with better examples.)