I have included the four translations below. In terms of 1 Corinthians 13, they are very close.
As a general background, the King James version took as its point of origin the pre-existing Tyndale and Geneva Bibles. The Geneva Bible is understood to be a more radical translation of the original Hebrew and Greek than the King James version. It was favored by the Protestants, and though banned in England, many copies were smuggled in.
King James wanted an authorized translation that would replace the Geneva Bible and use language more amenable to ideas of monarchy and hierarchy than the Geneva Bible.
The Douay-Rheims Bible was the Roman Catholic translation from the Latin rather than from Hebrew and Greek. It was translated in accord with Catholic theology and was most often used by English Catholics.
The Tyndale and Geneva versions are the translations that most favor the common person while the Douay-Rheims and the King James are more supportive of the privileged classes and the existing status quo. Often the differences in translation are subtle.
As for 1 Corinthians 13, the Tyndale and Geneva Bibles are most likely to use the word "love" while the other two translations use the word "charity." Other differences include, in verse 4, the Tyndale and Geneva versions describing love as "courteous" (Tyndale) and "bountiful," (Geneva) whereas the other two versions opt for the words "patient" and "kind." "Bountiful" carries a connotation of material generosity that "patient" and "kind" do not: we can understand how a more conservative version might move away from a "spread the wealth" translation.
As for the other verses, you can easily line up the translations below and compare them. As mentioned about, they are all very close:
Tyndale:
1 Though I spake with the tonges of me and angels and yet had no love I were eve as soundinge brasse: or as a tynklynge Cymball.
2And though I coulde prophesy and vnderstode all secretes and all knowledge: yee yf I had all fayth so that I coulde move moutayns oute of ther places and yet had no love I were nothynge.
3 And though I bestowed all my gooddes to fede ye poore and though I gave my body even that I burned and yet had no love it profeteth me nothinge.
4Love suffreth longe and is corteous. Love envieth not. Love doth not frowardly swelleth not dealeth
5 not dishonestly seketh not her awne is not provoked to anger thynketh not evyll
6 reioyseth not in iniquite: but reioyseth in ye trueth
7 suffreth all thynge beleveth allthynges hopeth all thynges endureth in all thynges.
8 Though that prophesyinge fayle other tonges shall cease ) or knowledge vanysshe awaye yet love falleth never awaye
9 For oure knowledge is vnparfect and oure prophesyinge is vnperfet.
10 But when yt which is parfect is come then yt which is vnparfet shall be done awaye.
11When I was a chylde I spake as a chylde I vnderstode as a childe I ymagened as a chylde. But assone as I was a man I put awaye childesshnes.
12 Now we se in a glasse even in a darke speakynge: but then shall we se face to face. Now I knowe vnparfectly: but then shall I knowe even as I am knowen.
13 Now abideth fayth hope and love even these thre: but the chefe of these is love.
Geneva:
1 Though I speak with the tongues of men and Angels, and have not love, I am as sounding brass, or a tinkling cymbal.
2 And though I had the gift of prophecy, and knew all secrets and all knowledge, yea, if I had all faith, so that I could remove mountains, and had not love, I were nothing.
3 And though I feed the poor with all my goods, and though I give my body, that I be burned, and have not love, it profiteth me nothing.
4 Love suffereth long: it is bountiful: love envieth not: love doth not boast itself: it is not puffed up:
5 It doth no uncomely thing: it seeketh not her own thing: it is not provoked to anger: it thinketh no evil:
6 It rejoiceth not in iniquity, but rejoiceth in the truth:
7 It suffereth all things: it believeth all things: it hopeth all things: it endureth all things.
8 Love doth never fall away, though that prophesyings be abolished, or the tongues cease, or knowledge vanish away.
9 For we know in part, and we prophesy in part.
10 But when that which is perfect, is come, then that which is in part shall be abolished.
11 When I was a child, I spake as a child: I understood as a child, I thought as a child: but when I became a man, I put away childish things.
12 For now we see through a glass darkly: but then shall we see face to face. Now I know in part: but then shall I know even as I am known.
13 And now abideth faith, hope and love, even these three: but the chiefest of these is love.
Douay-Rheims:
If I speak with the tongues of men, and of angels, and have not charity, I am become as sounding brass, or a tinkling cymbal. And if I should have prophecy and should know all mysteries, and all knowledge, and if I should have all faith, so that I could remove mountains, and have not charity, I am nothing. And if I should distribute all my goods to feed the poor, and if I should deliver my body to be burned, and have not charity, it profiteth me nothing. Charity is patient, is kind: charity envieth not, dealeth not perversely; is not puffed up; Is not ambitious, seeketh not her own, is not provoked to anger, thinketh no evil;
Rejoiceth not in iniquity, but rejoiceth with the truth; Beareth all things, believeth all things, hopeth all things, endureth all things. Charity never falleth away: whether prophecies shall be made void, or tongues shall cease, or knowledge shall be destroyed. For we know in part, and we prophesy in part. But when that which is perfect is come, that which is in part shall be done away.
When I was a child, I spoke as a child, I understood as a child, I thought as a child. But, when I became a man, I put away the things of a child. We see now through a glass in a dark manner; but then face to face. Now I know in part; but then I shall know even as I am known. And now there remain faith, hope, and charity, these three: but the greatest of these is charity.
King James:
1 Though I speak with the tongues of men and of angels, and have not charity, I am become as sounding brass, or a tinkling cymbal.
2 And though I have the gift of prophecy, and understand all mysteries, and all knowledge; and though I have all faith, so that I could remove mountains, and have not charity, I am nothing.
3 And though I bestow all my goods to feed the poor, and though I give my body to be burned, and have not charity, it profiteth me nothing.
4 Charity suffereth long, and is kind; charity envieth not; charity vaunteth not itself, is not puffed up,
5 Doth not behave itself unseemly, seeketh not her own, is not easily provoked, thinketh no evil;
6 Rejoiceth not in iniquity, but rejoiceth in the truth;
7 Beareth all things, believeth all things, hopeth all things, endureth all things.
8 Charity never faileth: but whether there be prophecies, they shall fail; whether there be tongues, they shall cease; whether there be knowledge, it shall vanish away.
9 For we know in part, and we prophesy in part.
10 But when that which is perfect is come, then that which is in part shall be done away.
11 When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things.
12 For now we see through a glass, darkly; but then face to face: now I know in part; but then shall I know even as also I am known.
13 And now abideth faith, hope, charity, these three; but the greatest of these is charity.