Having flown on different versions of both models, I consider they both have assets in different matters;
If we consider equivalent power plant (Arriel versions) on offshore ops.
S 76 offers a better payload ( about 300 lbs) on same distances....it usually flies slighly faster and provide more confort for passengers (specially when more than 8 ) It also requires more mechanical attention and a deeper system knowledge for pilots. I also believe it is less "forgiving" in marginal weather. The low blades on front sector is also something to keep in mind on prkings and helidecks.
Improvments like sperry DAFCS,EFIS,IDSS, external life rafts, central locking....make life much easier. This rotor aerodynamic is also wonderfull and behave like a freezbee....only 30 kts or so and we are quickly on the safe side of the heigh and velocity graph.
I feel the AS365 is more maneuvrable, more compact for confined or small helidecks, also simpler for pilots, maintenance and more confortable for crew. It behaves better in turbulent weather and is easier to configuration changes (cargo, medevac,sling) The too light doors and cowling design is not such a big hassle as long as we are aware of this.
To make it short, I'd choose the S76 for long sector ops (crew change on rigs, with few T/off and ldgs) since the Dauphin will be prefered as a "field" helicopter (short sector shuttling, many t/off ang ldgs, field maintenance facility)
I love both of them, and their plastic brains haven't shown any sign of jealousy untill now...(if only women were fitted with the same hardware !)