Variety makes some difference, but more important are weather, nurture and terroir. Our relatively cool, wet season this year isn't likely to encourage the most flavor development in herbs. Most herbs originate in hot, dry regions.
In my experience, herbs concentrate their essential oils best, reaching the highest levels of flavor, when they are a little stressed. If you grow them in fertile soil and feed and water them well, you get lush growth and, often, watered-down taste.
I've had the most success growing herbs in very well-drained plots. When I start a new herb bed, I select the driest, sunniest spot available, remove most of the indigenous soil and replace it with a mixture high in sand and low in nutrients. (Do not simply mix sand in with ordinary, heavy clay Chicagoland soil -- clay + sand = bricks!) Then I do not water unless the plants begin to look wilted.
I wouldn't recommend growing mint and basil together because they have opposite requirements. Basil is a Mediterranean herb requiring lots of sun, heat and dry conditions. Most mints prefer cool, lightly shady spots with a moist soil.