Originally posted on May 27, 2020.
A lot of my posts on AoPS are written in all lowercase, maybe with the occasional capitalized "I" or first word of a sentence in there. All of my essays are written with proper capitalization, punctuation, etc. Normally people dismiss these kind of differences as the product of laziness or because I want my essays to hold more weight. 
I think it'd be no surprise to those people that I usually write my essays with proper capitalization and grammar the first time. And I think it'd also not be surprising that I usually prune or rewrite sentences I don't like before I even write the next one. I also think it's no surprise that I write somewhat slowly, often pausing before I write the next word in the middle of a sentence. But I think the reason behind this would be at least enlightening, if not surprising.
You write when you have a bunch of thoughts in your head and you want to spill them onto paper, so you can actually see them and organize them. There are two ways this happens: You either have some subject you've been thinking about for a while and you're just carefully trying to sort out what you've realized, or exposure to a new idea or question leads to you typing out a response, only for you to realize you have more things to say. Because neither of these have names yet, let's call the first one "brewing" and the second one "flooding."
Brewing happens when there's some idea you've been thinking about, either consciously or subconsciously, for a while. When you're brewing it tends to be the more abstract and important questions you write about, or about opinions you have that you might be suppressing out of fear of controversy.  Questions like "how should I even go about doing math competitions," "why is it so hard to get started in math competitions," and "why should I write in the first place?" are the type of questions that you brew with.  Brewing is when you mostly know what you want to say, and you just want to sort it all out in front of you. It's like opening a shuffled deck of cards and ordering them.
Flooding is arguably the more interesting of the two, because it's what people realized for the first time. Most of the good that the forum does me isn't by presenting smart answers to smart questions - most of that ground has already been covered by other reputable users, and I tend to already know and agree with those answers. In fact, I think you realize things yourself when you see questions or statements so stupid that you feel the urge to go and say "What you just said was stupid, and here's why."  Flooding usually happens in response to something. Most surprising statements are floods, and surprising statements are usually the interesting ones. Flooding is when one train of thought leads to the next, and you're just throwing any good idea you have out there and connecting them together. It's like looking under your couch for spare change - you grab anything good you find and throw it out on the table, and make some attempt to sort it out later.
When people brew, they tend to write it carefully the first time. This means that there will be proper capitalization. When people flood, they'll tend to just throw words on the screen and maybe make a few edits so that the whole post actually sticks together. Brewing usually has capital letters, and flooding usually doesn't.  Go through the posts of any highly reputable user and you'll probably find the same pattern.  So, in fact, it's not a bad idea to write informally - if you're only writing formally, chances are you aren't noticing anything new, which is as red as a red flag can get. 
A couple of interesting things that I wanted to say but couldn't fit in the essay.
 After all, a post on an online forum does not need to be timeless and will not be re-read as much as say, something you save a link to on your website.
 As an example, take my thoughts on the AoPS Geometry book. At the time I bought the book my weakest subject was geometry. (This has reversed since then.) My gut reaction to the book was "Wait, this is boring and really sucks, why am I reading this?" Only three years later do I finally realize that I was right all along - and that a lot of the people who so zealously recommend the book do so because everybody else is, and not because they actually know what they're talking about. Don't suppress your gut reactions - even if you later change your mind at least it'll be clear to you why you think the way that you do.
 There are also specific questions that get brewed. The important thing is that it's that matters to you to some capacity, or it's a long term problem you've finally solved or figured out how to solve.
 Sometimes you'll realize that "bad" questions weren't actually bad. But I think surprisingly often, the instances of a lot of smart people realizing a lot of smart things are usually in response to a really stupid question. (Amusingly, the linked post doesn't fit the lowercase vs uppercase pattern - this post is probably one of my longest and most well-recieved floods.)
 This is probably just a common manefestation of a relaxed mental barrier. Unlike in flooding, where you don't quite know what you're trying to say and you're throwing out the first good idea that comes to mind, brewing is when you know what you want to say and you're just figuring out how to say it. For the former you need a relaxed mental barrier so your ideas can actually flow, and for the latter you have to have a high mental barrier so you can actually say what you mean, and do it well.
 Of course, moderators do not fit this pattern because everything they say has to look official. But I do think there are other tics out there that can distinguish this pattern. (Examples include "okay I feel like I should," "wait," "like," and "anyway," in that order. Other less good indicators are "you know" and "actually," and I actually think they're not so great because I tend to use them myself.) I also think that if you keep this in mind, it's pretty easy to distinguish between brewing and flooding.
 Don't take this too literally - your personal tics for brewing versus flooding might be different from mine. Just make some effort to figure out what those tics are, and make sure that you're at least occasionally flooding.