That would be a long answer full of different viewpoints. Many have used these words to describe music of their own or others, as though it can be nailed down, but no one seems to agree on the right way to divide them up. The same words can be used to describe very different traditions, and everyone is equally right I think.
I will take a shot as I understand it, simoplified due to tiredness! This is from a US view from one who does all these musics on a weekly basis, acoustically. And who has collected a large Gospel Tape Lending Library, which spans and transcends all of these.
I guess I should start with spirituals-- easiest. This usually refers to the music of African slaves and their descendants in the US before and after they were freed, religious and spiritual in nature and based on African call-response and other song forms. I have a delightful source book that describes this genre more fully and have been thinking about posting some of the editor's thoughts on that. Spirituals are generally understood to be inspired by feelings and events, an outpouring, songs to be sung in church, at work, out for a walk... like breathing, I think. Melody rules, over words, because words can be made up to suit the moment, and this method of improvising verses to make a new version or extend the theme of an existing version is called "floating verses." There are melodic features unique to the form also. Spirituals were passed mainly orally.
Hymns are generally understood to be be music composed on purpose for use in organized worship, written down and usually printed by denominationally-sponsored presses for authorized use in a particular denomination. Beyond that we get into all the kinds of hymns there are, and some, like Sacred Harp, are passed in an oral-written combination. But for the most part, if you are thinking hymns, you are thinking a hardcover book.
Gospel is used to describe the sub-genre of any other genre of music that is Christian in focus, and there are some, like bluegrass, that you might hear in a concert as part of an eclectic performance. There is Polka gospel, southern gospel, country gospel, blues gospel, etc. "Gospel" also means a number of other specific things depending on the tradition and view of the person saying it, and I will leave that to others to fill in. Sometimes gospel means white southern gospel, for instance, whcih has certain melodic and harmonic patterns and certain commonalities of imagery and theme. But to others, "gospel" is all black, a special kind of church singing. And it's ALL GOSPEL!
"Praise" music is a relatively recent genre. Its beginnings are variously said to have come from ex-potsmoking "Jesus freaks" new to Christianity or from Bill and Gloria Gaither's southern gospel music! As most people understand it, it consists of short, easily-sung items meant for the whole congregation to sing repetitiuously with a "praise and worship team" leading on various instruments and vocally. This term also has a lot of room in it for various meanings, but I think of it most closely with the Christian charismatic movement, because the idea for many is to lose oneself in the worship through music and be one ion the spirit. *G* And in some respects you could say that praise music is related to the singing of Psalms, although the music used would usually be quiote different.
Hope this is helpful, and as I said I am sure others have their own ways of saying these things.
But then there's Taize music as well, and other chant forms, and Bahamian chant praise singing...