That's comparing apples and oranges.

ANNOVAR is a tool for annotating genomic regions. It can annotate using RefSeq info as well as other databases that you mention such as SIFT or Polyphen.

SIFT, Polyphen, PROVEAN are tools to predict if a missense variant has an impact on protein function and can play a small role in deciding if a variant is of clinical significance (pathogenic). SIFT and Provean may additionally make predictions about insertions and deletions (indels).

ANNOVAR is a tool to annotate variants by different classes; eg [coding transcript, non coding transcript, intergenic] and [missense, nonsense, splice site, 5'UTR, 3'UTR, intronic, non-coding] and [frequency in general population] and [evolutionary conservation score of the nucleotide, GERP] and [physiochemical change in amino acid, Grantham score] etc. It queries many different data sets of such annotations. Annovar annotations can include SIFT, polyphen and provean if you wish.

As you say, LOFTEE is specific for annotation of loss of function variants.