Is there a significant difference between the two? As far as I can tell, they seem to be used interchangeably.
“On the principle” outpolls “under the principle” by pretty wide margins in published books scanned by Google Books, as shown in this graph. So odd did the latter phrase seem to me that I even threw in “under the principal”—which might well occur, for instance, in discussing the organization chart of an American public school, or in elaborating a taxonomy—as a phrase that you might have mistaken for it; but that too was used in this corpus far less than “on the principle.” My native-speaker sense and this statistical survey strongly agree in strongly preferring “on the principle.”
Martin Smith, however, cites in a comment the usage “under the principle of comity, one nation defers and gives effect to the laws and judicial decrees of another country” (from here). This usage does seem normal to me: I suspect it is fairly specific to legal and similar contexts, where the principles in question are rather normative than natural. I disagree with Martin, though, in thinking that on would do nicely here as well.
The use of under the principle implies some kind of governing rule. In other words there is a set of rules, which when adhered to, dictate that something must be the case or must be carried out in a particular way. The rule or principle governs what may or may not be done. Especially if these rules or principles are codified into some kind of written set of regulations, principles or laws, we can talk about things falling or not falling under certain established principles. We can also talk of what is permissible 'under' these principles. Under the principle of therefore often means under the jurisdiction of the principle:
Pro-choice people sometimes argue that the fetus, at least in the earlier stages of development, does not fall under the principle requiring the protection of human life.
This means that pro-choice people think that foetuses fall outside of the jurisdiction of this principle. Similarly the jurisdictions of the following principles are set out below:
Under the principle of international exhaustion, the first authorized sale of a product anywhere in the world exhausts the intellectual property rights associated with that product worldwide; under the principle of national exhaustion, the first sale exhausts the intellectual property rights in the country of sale only.
On the principle on the other hand is more often used to indicate the reason why something is or was done in a particular way. Whereas under the principle describes what must be done according to a particular rule, on the principle describes an underlying justification for an action, or even sometimes even just an organising principle. It is not so much that the principle has some authoritative moral or regulatory power over somebody, it is more that somebody is resorting to the principle as a means or organising or justifying their action. The principle may well not be codified or established:
Parking was allocated on the principle of first come first served.
The 'principle' of first come first served is hardly a principle at all except in the organising sense. There may well be better principles to allocate parking by, but this is the one that was resorted to. The following 'principle' is merely a rationale for a particular decision:
We had the party at 7.30 on the principle that it would be easier for people to get there after the rush hour.
Under the principle: under the jurisdiction of an established principle.
On the principle: the rationale or organising principle behind a course of action.