The quality of the soil (and thus availability of nutrients) has a real effect on the brix numbers of produce.  Higher degrees brix indicates more dissolved solids, including sugar content, and consequently sweeter produce.  Soil that has been farmed for many years is likely to be depleted in minor and trace elements, even if the farmer replaces the macro nutrients each season.  Garden soil tends to receive more care, plus usually doesn't have the added pressure of having been farmed for many generations so there is a greater availability of micro and trace nutrients.  That leads to healthier plants, producing more sugars which we notice in the form of tastier fruits and veggies.

All produce will benefit from better soil but I personally notice the difference vs. store bought most in things that aren't necessarily sweet.  For example, lettuce and peppers grown in my garden are noticeably sweeter than those bought in the store.  These are produce that even under optimal conditions have pretty low brix readings, so going from say 3 degrees brix to 6 degrees is a large jump.  Now, a garden grown water melon is probably sweeter than its store bought counter part but the difference is less noticeable to me because they are both pretty sweet anyway.  In the case of the water melon, quantifying the difference may very well require the use of a refractometer since the percentage change between say 12 degrees brix and 15 degrees brix is significantly less than in the previous example, even though the difference between the two is the same.

As someone else pointed out, ripeness at picking has an effect on the taste as well.  Farmers pick their produce before fully ripe because they need time to ship it and sell it before it goes bad.  Gardeners on the other hand can pick stuff when it is at its peak ripeness.

As to the Penn and Teller bit, I agree with their sentiment.  Organic is mostly a marketing ploy and you just ain't gonna be able to feed the world on organic kale.  Organic produce grown in marginal soil isn't going to taste as good as non-organic produce grown in soil with more nutrients available to the plant.  Likewise, produce grown in good, compost enriched, highly fertile soil will taste better than produce grown in poor soil, regardless of which is organic and which isn't.  The key is the soil, not some label.  Feed the soil, the soil will feed the plants, and the plants will feed you.  Hydro guys, bugger off, I'm not talking about you.

I will only eat asparagus out of my garden. The stuff in the stores here is old dried out crap.
When the season is over, it's over for me until next year.