I do not know PHAST so much, but as far I can read in the respective publications:

PHAST:

PHAST is not without some limitations. First, like all other database-driven annotation systems, PHAST obviously performs poorly at identifying novel phages, whose genes/proteins are not closely related to any record in the PHAST database. In this regard, the appearance of large numbers of proximal proteins with unknown function PHAST is not without some limitations. First, like all other database-driven annotation systems, PHAST obviously performs poorly at identifying novel phages, whose genes/proteins are not closely related to any record in the PHAST database. In this regard, the appearance of large numbers of proximal proteins with unknown function could be a good indication of a novel phage. could be a good indication of a novel phage.

PHISPY

PhiSpy was developed based on seven distinctive characteristics of prophages, i.e.protein length, transcription strand directionality,customized AT and GC skew, the abundance of unique phage words, phage insertion points and the similarity of phage proteins

So it seems that PHSPY consider also contextual genomic features and the identificaiton it is not only DB driven.