The short answer is that constitutive rules generally are more fundamental than the act. This is because the rules negotiate the social interaction to achieve the goal, whereas the illocutionary act is a tool to make that interaction occur smoothly.
Before answering in-depth, let's review:
A constitutive rule is one without which the social transaction cannot be met. The chess is the example used to show this. Thus, a necessary condition of a constitutive rule is that it is itself a necessary condition of the social construction of reality, in this case, the behavior understood as a chess game. A child can certainly move the pieces around on the board, but without participating in such a way that demonstrates an understanding of the constitutive rules, the child is not playing chess.
An illocutionary act is one that has at least two necessary conditions, one of which is that a conventional utterance is put forth, and the other that some sort of goal is realized, mainly an effect upon the hearer. This effect as per Austin is the perlocutionary act. Note, that a speaker's goal need not actually be realized. Let's say the adult playing chess wants to end the "game" (teaching experience, pseudo-game, etc.). So, she tells the child, "you've checkmated my king!". Since the child isn't playing by the constitutive rules of chess, the speaker's intent isn't so much as to make the literal (propositional) claim that checkmate has occurred according to the rules of chess, but rather that the activity is ending. Of course, the child wouldn't know this or what 'checkmate' means on a propositional or illocutionary level. Here the teacher is intending to familiarize the child with ultimate goal of chess by association. The child will begin to pair up the utterance "checkmate" with the end of the activity.
Now, let's compare the difference between constitutive rules and illocutionary acts to see how the two relate more properly.
In the social construction of reality, for two participants to be engaged in roughly the same reality (think chess game), both agents have to have essentially the same knowledge of the constitutive rules. Like a tyro and a grandmaster, that knowledge will not be identical, but it must be similar enough to allow the continued participation of both parties. It is on the basis of these constitutive rules that the illocutionary acts occur. Simply put, one cannot talk about and play a game of chess without knowing the rules of chess! Now here is the tricky part. *The necessity of the constitutive rules is domain-specific. If the chess teacher and the child start playing checkers, a much simpler board game, suddenly a child becomes capable of engaging in illocutionary acts since a simpler set of constitutive rules that the child is now familiar with is at play.
On page 40 of Speech Acts (Cambridge Press '87 reprint), Searle says:
Second, must there be rules (realized somehow) in order that it be possible to perform this or that illocutionary act? ... my answer to the second is that for most kinds of illocutionary acts, yes they are rule governed [sic]...
However, he does raise the possibility of his own error on the next page...
Even if it should turn out that I am wrong about question two, that illocutionary acts all can be performed standing outside any system of constitutive rules, it still would not follow that performing them in a language is not engaging a rule-governed form of behavior...
He's not too concerned, however, because his focus in this book (he has a follow up called the Social Construction of Reality where he doubles down and expands on his theory of speech acts) is on the speech act. To wit:
I hold both views, but it is only the answer to [another] question... which is crucial to my enterprise in this essay, because it is that view which articulates the hypothesis that speaking a language is engaging in a rule-governed form of behavior.