The quasi-binomial isn't necessarily a particular distribution; it describes a model for the relationship between variance and mean in generalized linear models which is $\phi$ times the variance for a binomial in terms of the mean for a binomial.
There is a distribution that fits such a specification (the obvious one - a scaled binomial), but that's not necessarily the aim when a quasi-binomial model is fitted; if you're fitting to data that's still 0-1 it can't be scaled binomial.
So the quasi-binomial variance model, via the $\phi$ parameter, can better deal with data for which the variance is larger (or, perhaps, smaller) than you'd get with binomial data, while not necessarily being an actual distribution at all.
When the response variable is a proportion (example values include 0.23, 0.11, 078, 0.98), a quasibinomial model will run in R but a binomial model will not
To my recollection a binomial model can be run in R with proportions*, but you have to have it set up right.
* there are three separate ways to give binomial data to R that I'm aware of. I am pretty sure that's one.