A contact created on the Android device is not synchronized to Thunderbird
When you create a contact in Android stock contact application, you can usually choose in which contact account you want to create it. It could be "Phone", "Google", "Facebook", etc. BirdieSync gives your the possibility to choose which Android contact accounts you wish to synchronize. Did you check that the account where you created the contact is selected to be synchronized in BirdieSync settings on the Android device ? You may access to these settings opening BirdieSync application, clicking on the "Settings" button and selecting the "Contact" tab.
For instance, if you created a contact in the Google account on your Androide device and if this account was not selected to be synchronized in BirdieSync settings, this new contact will not be synchronized. The solution is either to select the Google account to be synchronized in BirdieSync settings, or to create your contact in another synchronized account.