I'd say 'at times' is used more for states than intermittent activities:
'At times he seems happy, but at other times he sinks into a deep depression.'
To me, the other 3 expressions are closer in meaning to 'occasionally' but then there's such a lot of overlap between the two - one man's 'occasionally' might be another man's 'sometimes'. Which reminds me of a split-screen scene from Woody Allen's Annie Hall. A couple are seeing the same psychotherapist ( or 'shrink' as Woody prefers).
"How often do you have sex?"
He: "Hardly ever - maybe three times a week."
She: "Constantly! I'd say three times a week!"