Will a Count vectorizer ever perform (slightly) better than tf-idf? For the task of binary classification, I have a small data-set of a total 1000 texts (~590 positive and ~401 negative instances). With a training set of 800 and test set of 200, I get a (slightly) better accuracy for count vectorizer compared to the tf-idf. 

Additionally, count vectorizer picks out the relevant "words" training the model, while tf-idf does not pick those relevant words out. Even the confusion matrix for count vectorizer shows marginally better numbers compared to tf-idf. 

TFIDF confusion matrix
[[ 80  11]
 [  6 103]]
BoW confusion matrix
[[ 81  10]
 [  6 103]] 


I haven't tried cross-validation yet though it came to me as shock that count vectorizer performed a bit better than tfidf. Is it because my data set is too small or if I have't used any dimensionality reduction to reduce the number of words taken into account by both the classifiers. What is it that I am doing wrong?

I am sorry, if it is an immature question, but I am really new to ML.
I would say 1000 documents is a bit less to draw any conclusion about the vectorization technique, Neither an increase in True positive by 1 would matter. As the size of the vocabulary increases, TfidfVectorizer would be better able to differentiate rare words and commonly occurring words while Countvectorizer would still give equal weight to all words which is undesirable. So, TfidfVectorizer will give you better performance than CountVectorizer as the size of the vocabulary increases