In general, very few differences. For certain non-profits (e.g. 501(c)(3), churches) there may be exemptions from FUTA or SUTA or, in the case of SUTA for some states, a requirement to pay tax only when there is actual experience (employees receiving benefits). In certain situations there may be exemptions from FLSA minimum wage and overtime requirements (e.g. internships). Churches and clergy have special rules regarding FICA and SE taxes. This is not an exhaustive list.
There is also more of a "volunteer" issue with non-profits then for-profit firms. For example, I am an accountant. If I work at a non-profit as an accountant, I cannot also volunteer my time as an accountant. I run into the DOL MW/OT rules at that point. If however I have never been anything but a volunteer at a non-profit, it is (generally) not an issue.
Always remember that IRS and DOL have very different concerns and very different laws. Federal and state laws can also be different.
Also there may or may not be a tie back from wages/benefit costs back to the funding sources/grants, etc. This is HUGE at our nonprofit much like cost centers of a for profit. There is also a lot of reporting out that is done by the finance department which includes payroll.
We have very very few volunteers due to the community we serve (early childhood). But we do seem to have quite a few IC/ee issues that i have to constantly speak to managers about. But in the end, we run a lot like a for profit business.