Twitter API not showing old tweets?

You can use the REST API resource GET statuses/user_timeline to retrieve the most recent 3200 tweets from any public timeline.

I have the same problem as you, so after see that Twitter Web Search works I've started to implement my own solution, you can see on my GitHub. It is implemented in Java, but it will make a post on my blog to explain how to do in other languages. I've downloaded tweets without any problems, my last test I parse more than 600k within 2014 from some specific users.

Twitter doesn't return tweets older than a week through search api. Take a look at the limitations section from the below link:

https://dev.twitter.com/docs/using-search