Is tannin in cider important?
That rather depends on where you're coming from. Many traditional ciders such as those from Germany, Switzerland and the East of England have quite low levels of tannin. Most modern 'factory' ciders have rather little. But traditional ciders from NW France and SW England have noticeably higher levels, so the cider is markedly astringent to most people's taste, especially if it's 'dry' (unsweetened). That's because these areas have always used bittersweet and bittersharp apples which are characterised by high levels of tannin. The reasons for this are historical and not entirely clear. But if you want to make a traditional English West Country style of cider, you really have to use some high tannin apples. And, even if you don't, some tannin in a cider is highly desirable or it simply becomes too insipid for anyone's taste and lacks 'body'
http://www.cider.org.uk/tannin.htm