The most obvious disadvantage is that then voters don't know what their representatives are voting. Take Donald Trump for example. In his public statements during his presidential campaign, he's said that he opposes immigration outside the legally-defined methods. In private, he apparently told the New York Times editorial board not to listen to his public statements, that he didn't actually mean it. Previous public statements indicate that he was supportive of the bill Rubio helped write, although Trump disparages that bill now.
Why is this problematic? It encourages candidates to engage in inflammatory rhetoric. Trump at least could be held accountable if he reversed his behavior in office. It could hinder a reelection attempt. But for a legislator, their votes are everything. A legislator could publicly support legislation, anonymously vote against it, and then publicly denounce others for voting against it. This is fine for the legislators but hard on voters.
How would voters know for whom to vote? It's hard enough to guess how candidates will vote once in office. If votes were anonymous, then one couldn't even evaluate incumbent candidates. If you don't know who the best candidate is, then why vote? Publicly, every candidate would be in favor of whatever is popular. Privately, they'd behave differently. Turnout would likely drop precipitously, as voting has no effect.
This takes away the one great advantage of democracy, that everyone feels like they have a stake in the system. With this, we'd have essentially created an oligarchy of the best liars. People who wanted to change the system would have to resort to the old standby: violent revolution.
If this really interests you, there are systems in which it could work. The problem with it is that it makes voting for candidates not work. So stop selecting candidates via votes. Instead, randomly select a few thousand voters and draft them as legislators. Let their votes be anonymous. The downside of that of course is that the average citizen has no experience in writing or even reading legislation. Laws might be even worse written than they are now.