What major discrepancies exist between "versions" of the Bible? Based on a comment from @AndrewThompson, 


  'the' Bible (any one you care to name) is no more evidence of anything than the BOM is. You might take the numerous discrepancies between versions of 'the' Bible as the proof of that statement. 


there is a belief out there that "versions" of the Bible apparently have "numerous discrepancies".  I'd like to know what these are.

What is the most egregious textual variant measured in terms of its impact on theology?

P.S. If you are going to cite the game of "telephone" please be prepared to cite a specific example, demonstrably showing manuscript evidence. If there are so many "discrepancies" this should be an easy exercise. The Codex Vaticanus, the Codex Sinaiticus, and the Dead Sea Scrolls are all online.  Additionally, there are several "Critical Appartus"es, the best are by Bruce Metzger, that will catalog every known variation. These things can be checked, I'd just like a good example of something significant.
There are none.

Translation Choices aren't Biblical Version issues

I categorically have to rule out translation choices as different "versions" because by definition, languages that change will change their wording to make things clear. Whether one translates the term "cell phone" into German as Mobil Telefon or Handy, for instance, is completely arbitrary, although the act of translating does introduce different connotations. By the same token, should one literally translate Ich habe die Schnatuze voll damit! literally as "I have the nose full with it!" or idiomatically translate the meaning to "I'm sick of it" or "I'm fed up with it!"  Of course those expressions mean nothing until the language catches up with it, proving the point that language changes.

Unlike the Qu'ran, the Bible has no problem with itself being translated.  If you want to quibble over a translation, you are quibbling over language, not the "Bible."  As such, I would confine any "discrepancies" to the original Greek and Hebrew (and Aramaic parts of Daniel).

There are some very bad translations out there (see Was Jesus a Separate God?), but again, those are not the fault of the manuscripts, but rather translation choices that lacked scholarship.

Manuscripts are pretty consistent

There are admittedly, minor textual variations amongst the various manuscripts, but are there any that have any impact on theology?  Most are the grammatical equivalent of a transposed comma or a changed preposition. In a few cases (e.g. the Woman caught in adultery in John 8), there are sections that are missing in older manuscripts, but no textual variations once the story is added to the corpus.

In his book, The Text of the New Testament, It's Transmission, Corruption, and Restoration, Bruce Metzger analyzes thousands of manuscripts, and points out every miniscule variation amongst the miniscules, codices, papyri, and manuscripts.  Glancing through the linked table of contents, it will become obvious how minor the variations are.

When the Dead Sea Scrolls were discovered in the 1947, after having been buried for nearly two milennia, there were no significant discrepancies.

When Tischendorff uncovered (stole?) the Codex Sinaiticus in the 1800s, there were no significant discrepancies.

In short, this book is way less fluid than modern edits to Charles Dickens or William Shakespeare.



An examination of claims that manuscripts have significant theological variation
This is detail, but I think its worthwhile.  If you like the short answer, stop reading - but I think in trying to make the case that there is "significant theologcal variation, this text indavertently shows how little disagreement there really is

I'd like to show some examples that one commentator says are "significant", from @CwallenPoole's excellent answer:

Here is the case - 


  This statement is clearly false. It is not true to the evidence. Dr. Sumner wrote: "The rare parts about which there is still uncertainty do not effect [sic] in any way any doctrine." This is false! Doctrine IS affected. Dr. Robert L. Thomas, John MacArthur's professor in his California Seminary, wrote: "No major doctrine of scripture is affected by a variant reading." False, again. Dr. H.S. Miller wrote: "No doctrine is affected." False again. Dr. Stanley Gundry stated: "Only a few outstanding problems remain, and these do not affect doctrine or divine command to us." False again. Dr. Ernest Pickering wrote: "Important differences of textual readings are relatively few and almost none would affect any major Christian doctrine." False again!


N.B. That's a fair number of good scholars to disagree with.  (Kurt Aland is also mentioned, the same Aland of Nestle-Aland - probably the best Greek NT out there.)  So, we have a lot of scholars making the claim I'm making.  Let's give the author a chance to debunk them.



  John 3:15. "That whosoever believeth in him should not perish, but have eternal life."
  
  Do you know what the "B" (Vatican) and "Aleph" (Sinai) manuscripts do to the three words, "should not perish"? They REMOVE them. So, in the two false Greek texts, there's no hell in Jn. 3:15. What versions follow these corrupted Greek texts? The NIV follows them, the NASV follows them, and the NKJV in the footnotes, follows them. So do the other modern versions and perversions. For them, there is no hell in Jn. 3:15. Is this not a major doctrine?


So, the claims is "should not perish" is not supported by the manuscript evidence.  Okay, let's assume its true.  The doctrine of annihilationism is not predicated solely on this Scripture, and in any case, isn't accepted by most Christians.  In either case, the significant theological point is "Believe in Jesus, Have eternal life."  Hell has other scripture to make the case one way or the other.

  Mark 9:44 and 9:46. Another example is Mark 9:44 and 46. Both verses are gone: "Where their worm dieth not, and the fire is not quenched".
  
  Because "B" (Vatican) and "Aleph" (Sinai) remove both verses, so does the NKJV in the footnotes; so does the NASV (by putting them in brackets); and so does the NIV. So do the other modern versions and perversions. In so doing, they take away the fires of hell. Is this not a major doctrine? ... Now these false Egyptian Greek texts and the false English perversions will assist them in their heresy of a "fireless hell"!


Again, neither a widespread position, nor the exclusive basis of the claim.

  John 6:47. Let me see if you can accurately lead a soul to Christ using exclusively Jn. 6:47 as rendered in the new versions. Note John 6:47 in the KJV, where the Lord Jesus declared: "Verily, verily, I say unto you, He that believeth on me hath everlasting life."
  
  That verse is as clear as a bell, on how to receive "everlasting life." But, the Westcott and Hort Greek text, following the "B" (Vatican) and "Aleph" (Sinai) manuscripts, takes out those two vital and precious words, "on me." Because of their reliance on these false Egyptian Greek texts, the NIV also removes "on me." So does the NASV. So does the NKJV in the footnotes. So do the other modern versions and perversions. If you're trying to lead a soul to Christ with those new versions and perversions, using Jn. 6:47 exclusively, you'll never lead them to Christ, because "on me" (Christ) is gone from that verse in their perversions! All they say is something like this: "Whoever believes has everlasting life." Believes what? Their verse doesn't say. Their verse merely says "believes." According to these perversions of John 6:47, if I were to believe in atheism, Christ promises me everlasting life. The same if I believe in humanism, or in the Easter Bunny, or the Tooth Fairy, or in Santa Claus, or in Rudloph the Red-Nose Reindeer, or in Bugs Bunny, or in Buddhism, Taoism, Judaism, Modernism, or in anything else! That's major false doctrine in my judgement, and it stems directly from false Greek texts and false English perversions!


Okay, so we're missing the words "on me" from believes "on me".  From the whole of Scripture, does anybody really think John 6:47 allows belief in Santa Claus to get you into heaven?

  Romans 1:16. Here's what it says in the accurate KJV: "For I am not ashamed of the gospel of Christ: for it is the power of God unto salvation to every one that believeth; to the Jew first, and also to the Greek."
  
  The heretical Greek texts of "B" (Vatican) and "Aleph" (Sinai) remove the two words, "of Christ" in this verse. Because of this, the NIV also removes these words. So does the NASV. So does the NKJV in the footnotes. So do the other modern versions and perversions. This certainly is doctrine. "Gospel" means "good news" or a "good announcement." What "gospel" could be inserted there instead of the "gospel of Christ"? Was it the good news about a pay raise? Was it the good news about a new car, a new hat, or a new house? No! It's the gospel or good news about Christ. That's doctrine! That's theology!


So, is the "Gospel" so radically different from the "Gospel of Christ"?  Don't think its major.

  John 7:8. Was the Lord Jesus Christ a liar? If you believe the false Greek text, "Aleph" (Sinai), and some of the versions, He was. Note Jn. 7:8: "Go ye up unto this feast: I go not up yet unto this feast; for my time is not yet full come."
  
  According to the Greek text "Aleph" (Sinai), the word "yet" must be removed. The NASV omits it also. So does the NKJV in the footnotes. So do some other modern versions and perversions. Why do I say this removal of "yet" makes the Lord Jesus Christ out to be a liar? Because He went up to the feast in question. If He told his brethren that He was NOT going up to the feast, and then later went up to that feast, He would have told a lie, would He not? This certainly is a major theological doctrine. As in all of the other 356 doctrinal passages, the KJV has superior theology here. The perversions are inferior in their theology and doctrine! Stay away from them! 



Is it just me, or is this a stretech?

Here's the point - a scholar doing his best to present "theological differences" between manuscript versions fails to make a compelling case that theology is fundamentally altered by manuscript evidence.  I still challenge someone to show me a case where Bible "versions" really change theology dramatically.