Difference between packaged sentiment analysis tools (TextBlob/NLTK) and training your own classifier?
In general, training your own classifier is likely to perform better but it's going to take more time and effort:


Probably the NLTK system was trained on some generic data which might be very different from your target text. Since any supervised system assumes the same distribution between the training and test set, it's always better to train on a sample of your target data. 
By implementing your own method you can adapt it more specifically to your use case.


So it's a trade-off between the level of quality and the time you're ready to spend on annotating your own data and tuning your own classifier.