My gut feeling is that a handful of people is less than a few people in these contexts, but there is no real reason to believe that. Another person would feel differently. Honestly, you can't really say. They are both very vague terms in most contexts.
The only situation where I can see a clear distinction is where we are talking of 'relatively' few. If I invited 500,000 people to something and only 1000 turned up, I might say only a few of the people I invited came (relatively speaking). I wouldn't describe 1000 people as a handful.