Both are 25-item self-report questionnaires for tinnitus, both score from 0 to 100, and both are widely used. They measure slightly different things and are good at different jobs.
The Tinnitus Handicap Inventory (Newman et al., 1996) was the standard tinnitus questionnaire for over a decade. It asks about the disability tinnitus is causing across functional, emotional and catastrophic domains. It is good for a one-off severity classification (slight, mild, moderate, severe, catastrophic) and for comparing to historic clinical data.
The Tinnitus Functional Index (Meikle et al., 2012) was developed specifically to be sensitive to change over time. It has eight domains — intrusiveness, sense of control, cognition, sleep, hearing, relaxation, quality of life, and emotional distress — and is much better at picking up the meaningful but moderate improvements that treatment produces. If you are tracking yourself across weeks or months, TFI is the right tool. If you are getting a one-shot baseline at a clinic, THI is still common.
HushOS uses TFI-style items in the daily check-in because the goal is tracking change.