The US's policy is economically protectionist - it deprives one group at the benefit of another. In a free market, demand will always equal supply, no matter the free policies. The lowered supply of natural gas by ending the ban would raise natural gas prices, which would entice new people to enter the market, driving back up supply, and lowering prices back down to equilibrium.
In addition, one can always get something back from exporting natural gas, such as money. We are running a huge trade deficit, and exporting natural gas will not only help to boost the economy by increasing the production of it in the US, but it will also lower the trade deficit, thereby raising GDP on two accounts.
Natural gas is at a premium and our reliance on it is still strong. As such, we have no reason to export it when our own citizen can barely afford to keep up with the ever-rising rates. Exporting would bolster profit but leave even less supply for us, locally, thus further raising prices.
Although allowing the export of natural gas may result in short term economic growth, in the long term the effects would not be worth it for Americans. If we are shipping our natural gas to other countries the result will be less natural gas available for our citizens. More demand and less supply would lead to the price of natural gas rising for all Americans. Therefore, ending the ban on natural gas exports would hurt Americans financially more than it would help.
The U.S. only has but so much natural gas, or else it wouldn't be so expensive. And since it's so scarce, we just can't afford to send it elsewhere. What if we run out of our own supply? The process of removing it from the ground is so hazardous to the environment that we shouldn't extract any more of it than we absolutely have to.
A recent analysis by the DOE Energy Information Agency found that natural gas prices would rise domestically in proportion to how much is allowed to be exported because the price for LNG in the rest the world is consistently higher than in the US. Due to rising prices less natural gas would be used for manufacturing and electricity generation and more coal would be used. That will cost us in higher electric rates and fewer manufacturing jobs.