This has been on my mind far too long and I'm ready to spill it out.
Being trained to work in a writing center and dealing with different sorts of writers for a long time has given me a whole new outlook on the differences between typos and grammatical errors.
A typo is a typographical error. A synonym would be "misprint," which implies an accidental mistake. Example: I lauhged so hard. (It can be assumed the writer knows how to spell "laughed" correctly, but simply made an error in typing.)
A grammatical error is (in the simplest terms) the misuse of the mechanics of a language, often intentional but sometimes by mistake. Example: Where did you say your going? (One would generally assume the writer doesn't understand the difference between your and you're. However, sometimes a grammatical error can be a typo in disguise--the result of hasty hands or flying fingers.)
Anyone who has been trained to edit papers should know that both of these are the last things that should be checked. (I'm getting sassy here and I'm sorry. It's the writing tutor in me.) Instead, it's more helpful to guide a writer through things like focus, organization, development, and thesis before moving on to sentence-level errors, especially in a timed situation. If you start on the little things first, those sentences being corrected may no longer even be there when the writer has formed a second or third draft, and that time will have been wasted. Once the larger issues are hammered out, then it's time to nitpick and refine (as well as explain the rules as you're doing so).
Of course, that's all relative to academic situations. It's definitely necessary to help people polish their work as much as possible when it matters. Is it an assignment for a teacher? Yes, you should probably help. Is it an application to grad school? Yes, you should probably point out mistakes. Is it a bill that's gonna be read by all of US Congress and signed by the President? Yes, by all means, buff the shit out of that motherfucker.
Online, it's a totally different ball game. In a world where people start wars over their, there, and they're in comment sections, I'm constantly questioning the necessity of correction in various online mediums.
In general, I think correcting typos online is extremely tacky, and the person doing it usually has some sort of vengeful motivation for trying to make himself/herself seem high/mighty. By definition, typos are accidental. Correcting one is like explaining to someone they have an elephant sitting on their head--pretty pointless. They probably realize the elephant is there (because their head is crushed) and, if they don't, they'll probably catch it later on and feel silly about it.
Autocorrect is a both an angel from heaven (who helps us spell those tricky words like "diarrhea") and a demon straight from hell (who is always fucking around with the simplest of terms). This seems to be a source of many online typos, which is another reason correcting them seems so useless. My phone often changes "if" to "of" for no apparent reason. If I tweet, "I will die of I don't get this pizza inside me ASAP," one can assume I know the difference between "of" and "if" and, as long as you understand me, you should probably keep your mouth shut to avoid looking like a douche.
So, I've asked myself what I believe matters, and where.
Twitter? I vote a big fat NO for correcting both typos and grammatical errors. You've got 140 characters with which to express yourself. Heavens knows I always have to rearrange sentences in tricky, incorrect ways and use a few short form words in order to stay within the character limit. Tweets are short, hasty, and often thoughtless. Are my tweets graded? No. Is the President going to see my tweets? Probably not, although some of these fan girls who keep tweeting sexually explicit or violent things to their favorite pop stars might have a few government agents on their trails at some point. However, if I do have a close friend who keeps making the same grammatical errors over time (not typos), and I know they're making them in papers they're writing, I will occasionally let them know the rule via link.
Tumblr? I feel the same as I do about Twitter. People often make grammatical errors on purpose on Tumblr anyway for the sake of humor in text posts, so it's kinda pointless.
Facebook? I still feel the same as I do about Twitter. Facebook is rampant with people correcting both typos and grammatical errors, and it just seems old and tired and bitchy. If I had a friend making the same mistake over and over, and I knew they were writing something important, I might notify them via personal message. No need to be a public shamer.
Blogs? This is a toss-up for me, because blogs are typically meant for lengthier pieces of writing. I think the blog's intent should be analyzed before any corrections are submitted, although I still generally think correcting typos is tacky. If it's a political blog that's meant to be extremely professional, perhaps send a private message. Many news blogs will make corrections and note them at the bottom of entries. If it's a personal blog, I think correcting is generally pointless. If you've been reading the same person's blog for a while and have noticed a pattern of grammatical errors over time, perhaps send a private message or email if it's really bugging you that much. Let the blogger know you have no ill intentions, but be prepared for a negative response. People don't always take lightly to being corrected, and rightly so, if the medium doesn't call for it.
Many blogs ago, someone wrote me a message to let me know I was occasionally mixing up "use to" and "used to." In my case, these were typos. I know the difference between the two terms and their usages. However, in the mind of this person, these were grammatical errors and they needed to let me know. I didn't take offense to their message, mostly because I understood the intentions behind it and it was written in an inoffensive and nonaccusatory way.
In general, though, I don't think correcting a personal blog is necessary because I don't really proofread my own entries the same way I would if I were submitting an essay to a teacher, and I think people should understand that. I'm sure there are tons of cringe-worthy mistakes sprinkled throughout my entries. I'm sure there are tons of both typos and grammatical errors in this entry alone. I try to catch what I can, but I also don't really care because, as I've said many times, this is a personal blog and, once an entry is written, I've usually washed my hands of it and don't have any intentions of coming back and editing over and over. In an academic situation, you better believe I'd be hunched over this shit with a magnifying glass. Right now, the computer is telling me "nonaccusatory" isn't a word in that previous paragraph. Google is telling me something different. I've decided to use it anyway because this is my blog and it sounds right to me, but I would certainly find another word to use if this were for a professor. Perhaps I've even intentionally sprinkled some mistakes within this entry to see if you'll correct them. Perhaps this whole thing is just one big trap.
Spoken word in Internet videos? I think this should never be corrected. Spoken language is completely different than typed or written language. Ideas are expressed differently and words are used differently to get different messages across. Different, different, different.
After all is said and done, there could be an argument that how you present yourself online is a reflection of how you are in person (even with regards to how you write), and I would agree to a certain extent. People are going to judge your abilities based on what you do/say/type online, regardless of how correct or incorrect this is. I would, however, argue that this is not always right, and it would do the world a lot of good to recognize this. Some people tweet in all lower case letters. Obviously they know that upper case exists, but they choose to type in lower case because they're either simply lazy with their typing, or else they're trying to convey a certain tone, or else they simply don't understand their keyboard. I fully believe Cher* knows how to write (and possibly quite well) but, reading her tweets, one would think she's never passed second grade. I don't believe we should be assuming those sorts of things about people based on their online interactions. Is everyone going to agree with me on this? No, and that's fine.
I've even been guilty of being that public shamer online--that douche-y typo and grammar warrior out for blood and wielding my Blade of Your and my Shield of There--and it's never made me feel very good about myself. Feel free to call me out on those past indiscretions. They won't be happening anymore. I'm sorry.
Overall, I think we could all stand to be a little more accepting and tolerant of both typos and grammatical errors--and this is coming from a writing graduate, former writing tutor, and self-proclaimed grammar Nazi. If you can understand what someone is trying to say online, I rarely believe there's a necessary time to harp on them. If you feel you must correct something, be considerate and send the person a private message. Most of us feel silly and ashamed when we realize our mistakes are out there (especially when we can't take them back), so softening that blow is always welcome.
Most importantly, keep writing! I don't think anyone should be discouraged for building their writing skills and trying their best to get a message out there. Once we understand this, we open up our minds to all sorts of new ideas we may have previously shot down and missed out on. There may be a really interesting fanfic out there about Hillary Clinton's pantsuits or a Big Bird/Snuffleupagus slash or a One Direction orgy that you've missed out on because of a few typos or grammatical errors that threw you off. Well, it's time, my friend. It's time.
*Edit: I didn't know Cher was dyslexic when I wrote this so I probably sound like a douche making fun of someone with dyslexia. I'd change it, but enough people have already seen it that it wouldn't make sense to. Just know those weren't my intentions, but I'm sorry regardless.