There is no such distinction generally, though there are geographical patterns, and personal favourings of one over the other in different cases.
As Janus notes in his answer, some people have objected to the past-tense form plead and pled for past and participle. We can imagine that these are people more used to the form pleaded for all cases.
I disagree with his reading of "both forms have often been associated with legal usage" as meaning that legal senses are treated differently than others. While there are words and senses (including some senses of plead) that are particularly associated with law, this is not quite the case here as the sense of plead in terms of entering a plea of guilty, non-guilty or nolo contendere is a term with which the layman is well-acquainted.
Rather, the fact that both have "often been associated with legal usage" means that we have strong evidence of both being much used in the language, and also while the OED does not claim to have a basis to standardise the language, the courts do when the terms are of particular importance to them, and so we can argue both approaches to the past tense to be "standard" at least by one "standard", and that one with an influence upon the language, and this then is part of what the lexicographer notes.
Also, "he pleaded with the king for mercy" is a legal sense. "He pleaded with the psycho-sexual killer" would make it non-legal; kings have a very different standing under the law, as does pleading with them.
In the examples that same dictionary gives, we can find both form used for sense particularly associated with law or other petitioning of authorities, and for those senses which are not.
Pleaded predominates in all senses, so I'll just quote "1941 E. R. Eddison Fish Dinner vii. 103 Should a been unlorded long since,..but the Vicar pled for him." as an example of pled in a non-legal sense.
Now, when we have different past-tense forms of a word, one sometimes displaces the other (though here both forms are old and still reasonably strong) and just sometimes they split in meaning.
Stricken and struck now mean different things (though there is an overlap of senses where either can be used) and there is the rather arbitrary hanged / hung distinction where "meat is hung, men are hanged"* that quite likely exists for no reason other than people have enjoyed pointing out other people were using the terms "incorrectly" for centuries, and so a distinction that might well have blurred naturally as the language evolved has continued.
These distinctions no doubt arose from someone like you having a sense that one form matched one sense better than another. It's interesting that in a question here about the difference between lit and lighted (there is none) and why Hemmingway favoured the less-common lighted (it was then more common, and also it rhymes with other words he was using) there were a few different personal theories about the difference between them, none of which have any evidence descriptively, but all of which are perfectly fine in terms of how someone might choose to pick between them (the prescriptive can actually be more liberal than the descriptive, when one prescribes only for oneself).
Your associating one past-tense of plead with one sense, and the other with another is another example of this; it's not backed up by records of usage, which is much more distributed by geography than the sense, but it's a reasonable basis to pick when choosing for yourself.