Simply put, the characters from your old characterset don't "hold the same place" as they do in the new (here is an example showing the placement of different values in the Windows 1252 characterset), hence why you see the artifacts.

You could find the relation of the different artifacts and find/replace in the database OR activetext, or you could just live with it like countless other communities have done after converting.

Depending on your character set and the language used on your forums (Asian languages for example use quite a bit of non-Latin characters) there may be a lot more items than I'm including in the below list; but some common characters are:
“ = left quote = “
” = right quote = ”
‘ = left single quote = ‘
’ = right single quote = ’
— = en dash = –
– = em dash = —
• = hyphen = -
… = ellipsis = …

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