Bournemouth certainly has 'the edge' when it comes to leisure travel, but most of Southampton's business travellers (previously using the former Flybe services to the likes of Amsterdam, Glasgow, Edinburgh, Manchester, Belfast, Paris etc would probably never relocate to Bournemouth.
A fair proportion of this business is/was generated from areas further north and east of Southampton. The journey across the New Forest to Bournemouth can be just as long time-wise as the drive to LHR, so much of this business would simply either relocate to Heathrow or, in the case of domestic traffic, cease to exist with people electing to drive - which is probably even more likely.
Bournemouth however, with better public transport connections, would absolutely thrive with leisure/city-break and holiday/sun-route business.