He swam slowly to the island.
He slowly swam to the island.
Some experts say that there is a “slight difference” in meaning.
Would you please tell me that difference?
No. There is no difference in meaning. Manner adverbs like slowly can be slotted into a number of niches, depending, I suspect, more on how the speaker wants the sentence to sound, in terms of rhythm and intonation, than anything else.
However, this is not to say that those with their own view of proper language use might not believe that there is some such meaning difference, and that they understand what it is. And, as a public service, they will often tell you about this.
Since such individual views of proper language are all imaginary, they are individually quite different and often contradictory; this means that all that purposeful usage cancels out in the long run.
But, in the short run, anyone who gets used to the way some such person talks gets accustomed to such usages, and often comes to believe that they're universal. If enough people do so, they might in fact become universal. But there can be a lot of competition and pushback from people who don't talk that way. This is the ordinary way language changes.
These adverb positions aren't always completely interchangeable.
If you're explaining the reason for the slowness of his swimming, the adverb must come last, especially in writing. In the second example below, when speaking, you might be able to convey this fact by emphasizing slowly (but you probably wouldn't actually do this unless "to the island" were replaced by a much longer phrase). The third is wrong if you're trying to say he was swimming slowly because his leg was injured.
Because his leg was injured, he swam to the island slowly.
?Because his leg was injured, he swam slowly to the island.
*Because his leg was injured, he slowly swam to the island.
When an adverb has a dual existance as a pragmatic marker (some would say a sentence adverb / adverbial), care has to be taken:
Clearly, he saw what was happening in the room. (pragmatic [modal] marker)
He clearly saw what was happening in the room. (ambiguous)
He saw clearly what was happening in the room. (manner [or is it degree?] adverb)
In Peter's examples, the third example (Because his leg was injured, he slowly swam to the island.) is grammatical with Because his leg was injured the reason for his swimming not his slowness.