I've ridden with both types and the convenience and accessibility of STI/Ergo integrated "brifters" is so much better than barends that there is no comparison in my experience. The hillier the terrain, the bigger the advantage brifters have. You can shift brifters while standing and barends are nearly impossible to reach from that position.
Barends are less likely to give mechanical problems. They aren't immune and brifters have gotten very reliable over the past few years but barends do have friction as a fall back option. They are also MUCH cheaper so manufacturers will use them to reduce costs if possible.
BTW, there is one other option; Kelly Take Offs. These are brackets that fasten under standard brake levers and mount downtube shifters just inboard of the brake levers. They put the levers in reach of your thumbs while riding on the hoods and are nearly (but not quite) as convenient as brifters. Since downtube levers all have a friction option, you have that fall-back too. They sell for $50 to $70 and use any model or "speed" downtube shifters so the entire thing is much less than any brifter. Here is a reference: http://kellybike.com/1st_rrwhatnot_archive1.html
I have a set of Take Offs on one bike and I'm quire fond of them They are far better than barend or regular downtube shifters at a fraction the cost of brifters.