1. The outdated statistics cause generating an inaccurate estimated plan. The estimated execution plan is created by Query Optimizer which uses both statistics and the query processor tree.
2. The actual execution plan is created after executing the query. When the stats are stale, the Storage Engine generates its own plan which is different than the estimated one.
The question: Why to bother about outdated stats if SQL Server can create accurate actual execution plan (by Storage Engine) when it finds out that data has been changed? How can the difference between the estimated and actual plan influences the query performance?