Which textual variants of the Greek New Testament are important? Well, it really depends on who is asking, and why you are asking the question. Regardless, it is a good question.
The Variant
Let us take John 6:44 for a test case, specifically the variant addition of μου after πατηρ. To find this very important variant I flipped open my NA27 and picked a variant on the page. So since providence has picked this variant, we will discuss it :)
What does this addition of μου do to the text? Well, it changes the translation from "No man can come to me, except the Father which hath sent me draw him: and I will raise him up at the last day." to "No man can come to me, except my Father which hath sent me draw him: and I will raise him up at the last day." According to the apparatus this variant has one papyrus and a few other unspecified manuscripts for support, so its likelihood is less than stellar.
Despite this, or perhaps because of its poor testimony and its rather obvious non-significance for the meaning of the passage, it makes for a great example. Who you are makes this variant important or not.
The Hapless Victim of My Blog Post
If great Aunt Hilda of First Fundamentalist Church of Dallas was sitting in her pew on Sunday, reading her KJV while listening to the preacher who talked about this variant and said "I think this text should read 'my father', not 'the father'", it is quite possible that great Aunt Hilda is going to have a fit. Chances are she thinks God was holding the pen of the apostles when they were writing, and was holding the pens of the translators of the KJV and kept them from making a mistake here. What she is reading is, after all, the Word of God. To her, the preacher has just added to the unchanging Word of God and has done something that is just unforgiveable. She might just go up to the preacher after the service and quote Rev 22:18 to this heretic preacher: "For I testify unto every man that heareth the words of the prophecy of this book, If any man shall add unto these things, God shall add unto him the plagues that are written in this book." She will then talk to the deacons and after a quick heresy trial the pastor will be removed and they will be satisfied that they have preserved the truth for all of those around them.
But I am not like her. And if you are reading this blog, you probably are not either. But she exists. Maybe her name isn't "Hilda" and sometimes she's a guy, but there are a lot of those out there. To them this variant is of great importance (if someone takes the time to bring it up), despite the fact that the variant simply specifies what is already obvious in the meaning of the text.
So What Is The Question Again?
So the real question should be the following: Which variants should we think are important? That's the first question that needs to be asked, but we can't answer it yet, because we have to ask a follow-up question. For what purpose are we asking?
This purpose is quite important. I can both say that this variant is important and yet say that it is of practically no importance. It is important because finding out what the text originally said, as far as can be discerned, is a valuable exercise. Let's say you get the text edited for the Gospel of John well enough to get the general meaning, but you are shaky on hundreds of particulars like implied (or not) possessives or omitted (or not) articles. How are you going to do stylistic analysis of the work? How about discourse analysis? What about syntax? How will you hope to compare the syntax of the Pericope Adulterae against the rest of the book if you don't understand the particulars of John's style? So for someone like me who is very interested in that sort of thing, yes, μου is quite important. However, how does this variant affect my understanding of the passage? How does it affect my view of Christ? Does my salvation hang on the interpretation of this verse based on the presence or absence of this possessive? It doesn't, it doesn't, and no. So is this variant important in that sense? Not at all.
So which variants should you think are important? It depends on your purpose. Should this one be important to the man in the pew? Not really. To the researcher? Sure is.
What Are the Character of the Variations in the Greek New Testament Textual Tradition?
In all likelihood, if you are reading this post, this is why you made it this far. Are the variants in the textual tradition important? Should the average Joe worry about our text or should he be confident and chill out.
From everything I can see, he should chill. What should be ultimately significant for a person of faith is this: Are there any viable variants that call into question Christian orthodoxy? From everything I have seen and heard, the answer to that is "no". And, frankly, variants that would affect orthodoxy but aren't viable are very hard to come by too, much less viable ones! I can't think of a single one. That's not to say they aren't there...they are just not the norm.
Most of the variants I have seen fall in the category which, for those purposes, are completely irrelevant. These include but are not limited to variations in spelling, omissions or additions of words, re-orderings and substitutions of synonyms. Only omissions and additions would generally affect translation, and then not in any way that would make me think "You know, salvation isn't by grace" or "Based on this variant I no longer believe in the deity of Christ." There are some variations for or against orthodoxy, but they are rare. If the question is "What does the New Testament have to say about God, Jesus, and Salvation", then textual criticism should rarely enter into the discussion.
There are some, though, that do affect the interpretation of a passage. Those are important, but not as common as you might think. Look in your textual apparatus and you will find that those are rare too.
Distrust the Manuscripts of the New Testament or Not? That Is the Question.
In brief, yes. Do we have as much evidence as we would like? No. Does the evidence we have point toward reliability or variability? From what I can see, the former. Hope that helps.

Brett
Eric
No need to be sorry. It was a good question.
Brett
Brett
Chuck Grantham
Eric
No problem previewing for me. Haven't heard from anyone else.
I don't know of anything like that with the NT. Some suggestions for conjectural emendation (i.e., changes suggested without actual textual evidence) have been made, but I know of none that have gained acceptance.
Wieland Willker
a) Codex Bezae has some large lacunae in Acts.
b) Codex Bezae reads at certain points in Acts alone the correct text (if one accepts NA).
From this it can be deduced that Codex Bezae also probably read in the missing parts at times alone the correct text.
This text is lost.
Eric
Wieland: Well that's something I didn't know. Do you know of a good writeup on that which I could read?
Brett
Wieland Willker
Ok, I give an example. Let's say, we have a manuscript Codex Bretianus.
It is extant in Acts ch. 1-14 only. From a careful analysis the editors of NA decide that it reads the correct text in 7 passages alone. That means, no other MS reads thus.
Applying simple statistics one can deduce that, if the MS would be extant in Acts ch. 15-28 also, it would probably read there the correct text alone also about 7 times.
But since this part of the MS is lost, we have lost the original text at 7 passages.
Now check NA for how often Codex Bezae reads the correct text alone in Acts and calculate ...
Eric: I don't remember where exactly I read this argument first, it was probably in some TC intro book.
Brett
Wieland Willker
Chuck Grantham
Eric
Brett: I think the Acts example is close, but the 2 Peter example does seem to be of the same character.
The Luke 4:44 example is interesting. Textually, the choice made by the NET folks is a good one. It is hard to see how the reading of Judea would have arisen out of the other two, but the other two could have quite easily arisen out of the Judea reading.
Is it an error? It is a little hard to see that it is not, but I have not checked the commentaries on it.
Wieland: Nice example. I haven't heard of that one. Is that written down anywhere yet or were you just chatting with the committee members?
Brett
Wieland Willker
Eric
Thanks. I've got the volume for James. I need to pick up what else is available...