I try and grow heirlooms almost exclusively. I like the taste and the idea that these are not genetically engineered to produce more or be more disease resistant. But, I am not saying there is anything wrong with hybrids. I grow some hybrid flowers in my gardens, and they are beautiful.
I think it is a personal choice. I love antiques. I go through thrift stores, yard sales, etc. always looking for a forgotten treasure. I see heirloom plants the same way. My MIL and I will go around to abandoned homesites and dig up old bulbs, and things like that. I now have a variety of daffodil that is not in any bulb book. I had a past president of the daffodil growers association look at a pic and he couldn't identify it. It is a lost variety, and is precious to me. I will eventually send some to him and let him grow them. And it makes me feel good to pass on a variety of my favorite flower that could very well have gone extinct!
It's no different than a heritage breed of poultry vs. red star/black star/cornish cross, etc. It's all a matter of personal preference.
I grow only heirloom tomatoes. My neighbors wait for them to ripen and we have a little garden party with an heirloom tomato salad. It's a lot of fun, and they get to try varieties they never would have before.
The thing about heirlooms is that your location is important. They will not flourish if your growing conditions are not similar to their home country. At least not without a lot of work! I highly suggest Baker Creek that other people have mentioned. I've been ordering from them for years and they are wonderful. In the plant description it will tell youif they are good for the Northwest, South, dry areas, moist, etc. Pay attention to those tips...it can save you a lot of time and heartache.
Shelly