You can find a pretty good analysis of the differences here:
According to the Arabic usage, fuqara are all those people who
depend on others for the necessities of life. The Arabic word fuqara
is a general word for all those who are needy because of some physical
defects, or old age, or temporarily out of living means, and can
become self-supporting if they are helped like orphans, widows, the
unemployed, etc.
According to the Arabic usage, masakin are those indigent people
who are in greater distress than the needy people usually are. The
Prophet (peace be upon him) especially enjoined the Muslims to help
such people as are unable to find the necessary means to satisfy their
wants and are in very straitened circumstances, but are so
self-respecting that they would not beg for anything, nor would others
judge from their outward appearance that they were deserving people.
According to a tradition miskin is one who cannot make ends meet,
though his appearance does not show that he needs help nor does he beg
for help. In short, he is a self-respecting person who has become
needy.
The descriptions are also here, in the section "Who is a poor person and is defined as a pauper":
A Faqir is a person who has some property but not enough to make
the qualifying threshold known as Nisaab or he has enough to fulfill
the Nisaab but some of or all of the property is part of his basic
necessities, such as a house to live in and clothes to wear and
servants for his care and tools for his profession, then regardless of
how expensive they are, they are not counted in Nisaab and if his
savings do no total the Nisaab or he has savings but he is in debt and
when his debt is calculated it takes his savings below the Nisaab
threshold, then this person is regarded as a poor person.
A Misqueen is a person who has nothing and is desperate even for
shelter or for clothing to cover his body and has to resort to
begging. It is allowed for a Misqueen to beg and it is not allowed for
a Faqir to beg. This is because if a person has enough for food or
clothing to cover themselves then it is Haram for a person to beg.
As far as I can tell, all categories are treated equally with regards to sadaqah/zakat, unless they are relatives, in which case they generally get preferential treatment (and it is not called charity, to save their pride).