A bunch of random thoughts about image display, in which I often contradict myself.
The advantage to online pics is that I already have a web site with gallery software, and I tend to upload almost all of my pictures there. The gallery software has an automatic resize function, so I generally just upload the raw pics. If I use the inline function, I can just link to my the picture in my gallery and my work is done. In order to send the picture as an attachment, I have to first save it to my local computer, do further resizing if necessary, then upload it to TBN.
On the other hand, most people do not have that kind of gallery setup, and setting up pictures for linking ranges anywhere from difficult to impossible. For these folks, the attachment feature is better than sliced bread. If you use the attachment feature, it takes additional steps to then link to your own uploaded picture and show it as an inline image. I've never iunderstood why anyone would go to all that extra work.
Even though it's easier for me to use linked inline images, I still remember my dialup days and go to the extra trouble to use attachments.
One other advantage to inline images is that I can display an image, write some text, display another image, write another caption, and so on, all in one post, instead of making multiple posts.
Most forums require linking to pictures because they are not as generous as TBN to provide storage space for all those pictures.