If you have extra pages for your website like a forum, which do you think is better to use between these domains?

extra,website,com

www,website,com/extra

I changed the dot (.) into comma (,) so it won't link to a website which I don't know.

What are the differences between these two?

Well, I'm not exactly sure of the implications, but here are the differences that stick out:

Subdomain
-It has a DNS entry
-Data is at the root of the website
-Can have separate robots.txt files

Subdirectory
-No DNS
-Sometimes complicates installs (most forums compensate for this)
-Robots/Spiders have to dig deeper in your site when crawling

Interesting question. Hopefully, some others will chime in on this.

Personally, I prefer the subdomain method, and the reason is that on a subdomain I can pop that on a seperate server and just link the IP numbers rather than having to do rewrites and redirects on a folder method.

The advantage is that you can then spread your sites (or empire) over multiple machines which you can tailor to the needs of that task specifically. If your forum gets too popular, you just need to upgrade THAT server (or add more memory etc) rather than having to worry about moving ALL things.

The other item is that by spreading it over multiple machines, you're giving your users or clients multiple ways to get a hold of you should one of your servers go offline. This goes back to the old adage of "don't put all your eggs in one basket"

If you do decide to create a forum in a subdomain, make a 301 redirect on a folder for good measure

I prefer using subdomain because sometimes using directories complicates the process of installing scripts.

Agree! Directories are sometimes a big headache!

So if directories are a headache then why is it that there are still many people who are doing it instead of sub-domains? Is it because they don't know how to make a sub-domain or are they just used to creating sub-directories?

Directories have their place. Some hosts don't allow subdomains. Some users don't like to have accounts on separate machines. It's really all what you make of it. There's no right or wrong way of doing things. You'll see places like Verizon or AT&T that say go to "xxxx.domain.com" for certain promos and then other times they say go to "domain.com/xxxx" - it's all just a matter of how they're promoting and what SOUNDS good to the visitor in many cases.

I am agree with Hesham A.Megid & arbet.

Obviously subdomain is good because it has separate entity.