A function query is a query that invokes a function on one (or more) of the fields available. You add a function query if the value you have in a field has to be processed to get the value you want - just as you'd do in a mathematical sense.
Showing "the difference between a query with function queries and without" isn't really possible, as they don't do the same thing. You pick one (or both) depending on what you need.
An adopted example from the reference manual - Lets imagine we have a set of documents that describe users, and these users have two fields - mails_read and mails_received. To get anyone that has read less than 50% of their mails, we can apply a filter query as a function (with the frange query parser) (fq here means filter query - the frange is what makes it a function query):
fq={!frange l=0 u=0.5}div(mails_read,mails_received)
Otherwise we'd be limited to receive those who just had read a specific range of emails or that had received a specific range of emails - or we'd have to index a value that kept the updated value for mails_read / mails_received each time we updated the document (which is a perfectly valid strategy, and usually more efficient).
Another example is to use a function query for boosting documents, and the most common one is to boost by recency (i.e. that a more recent document receives a larger boost):
bf=recip(ms(NOW/HOUR,mydatefield),3.16e-11,1,1)
This applies the recip function to the difference (expressed in milliseconds) between the mydatefield field and the current hour.
recip: Performs a reciprocal function with recip(x,m,a,b) implementing a/(m*x+b) where m,a,b are constants, and x is any arbitrarily complex function.
Yet another fine use case is to use the special _val_ field - if you query against this magic field with a function, the value returned by the function will be used as the score of the document (instead of affecting it through boosting or limiting the resulting set of documents as a query).
_val_:"div(popularity, price)"
.. would give the score of the document based on the result of the division (what the values represent is up to you).