Polling is far more opaque and highly susceptible to corruption. Any pollster worth his or her salt can get exactly the outcome desired by tweaking the protocol and consumers of the poll results are none the wiser.
I'd almost go so far as to say that polls are garbage and shouldn't even exist, let alone replace voting, but that may be going too far.
If polls are garbage then voting is even more garbage. At least polling can use statistical sampling to esimate the true result. For the expense of elections and polling and paying for all the campaigns and media and hoopla you could have incredible amount of polling, from many organizations, and figure out what people actualy want.
As it is, no matter how many people want eg certain gun control regulations, or not to go to war with a country, it will never happen due to special interests.
It is far harder to fool all the people all the time than to buy off and control some of the people all the time.
I'd almost go so far as to say that polls are garbage and shouldn't even exist, let alone replace voting, but that may be going too far.