Wikipedia indicates an heirloom plant "is a cultivar that was commonly grown during earlier periods in human history, but which is not used in modern large-scale agriculture."
What we call heirlooms plants were the standard years ago. Brandywine tomatoes were common 50 to 75 years ago. One hundred forty years ago, sweet corn was planted and harvested.
The difference is that somewhere between then and now, we have lost many varieties...many. Here's a graph a blog on extension.org which tells the story well:
What you see is the change in the number of varieties from 1903 to 1983 in each type plant versus. We have lost many varieties, never to come back again. They are extinct.
Existing heirlooms are making somewhat of a rise starting in farmers markets and now in some grocery stores. Some are unusual looking. If all you have seen are round red tomatoes, you might not know what to make of green zebra tomatoes.
Professional growers like predictable product they can sell. Tomatoes, for example, are usually picked green and gassed to make them red. Thus their flavor is often not as good as heirlooms you might buy at a farmers market.
Heirlooms are important because they are a part of heritage to pass down. If it's not preserved and valued, the only food available is what producers and growers find most reliable and easiest.