The difference is that redo does not re-evaluate the loop condition (in this case: "(<DATA>)", which fetches the next line) before evaluating the loop body again, whereas next does.
This is why in jwkrahn's solution, the next line is fetched manually before calling redo:
$_ .= <$fh>;
The advantage of jwkrahn's solution with redo, is that the implicit variable $_ can be used to store the complete multiline record.
The advantage of kcott's solution with next, is that there is only one place where the <> operator for fetching the next line is used (inside the loop condition) - but re-evaluating the the loop condition also resets $_, so in this case a custom variable needs to be declared above the loop to store the current record.