Just like indexes any additional statistics create maintenance overhead in both space used (it has to store it someplace) and the duration of maintenance (it does take time to update them.) If auto update stats is on then that can lead to slowness during the day while SQL updates stats that it may not even be using. If it's not on that means time needs to be put into developing a maintenance plan that will do that and a longer duration for maintenance. I would have to defer to someone with more knowledge on this point but I would also think that with additional statistics to consider when calculating the query plan compiling plans can take longer.

There are also columns where creating stats will never have the opportunity to be used. If SQL isn't using that column to search then the stats won't be used. And if a column is frequently being used but isn't indexed and doesn't have a manually created stat on it then SQL can create one if auto create stats is on. We have created stats in our DB but for the most part we rely on SQL to do that when needed.