There are two things to be aware of about Contacts search in iOS:
- If you search from Contacts.app iOS will search both person name and business name.
- If you search from global (spotlight) search iOS will only search one of these, depending on Contact type.
This is sufficiently weird and inconsistent that it may qualify as a bug.
In general search on iOS is "annoying". PalmOS did a much better job in 1997.