We don't know anything about Snape's wand. What sort of wood? The core and length? For info on known wands for comparison, here is a link to the Lexicon wand page.
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Re: Secondhand wands
I think it's an intensified version of what Severus told his DADA class about nonverbal magic--"It's a question of concentration and mind power, which some ... lack." We see lots of people doing nonverbal spells, eventually; that I recall, we only see Riddle, Snape, and Dumbledore perform wandless magic as adults.
It does seem to be a question of focus, rather than power per se. And the involuntary magic of kids seems to be mostly along the lines of countering immediate threats (and to operate in an inconveniently literal manner).