Rather than looking how it works when we try to solve puzzles, one of the difference is also in lifestyle.
As Ti is like a big frame that continues to grow, it is absolutely imperative we store information in that frame as truthfully as possible. If we become half-assed and start to assume certain things without giving them any thought, that big frame we've built for ourselves is going to crumble. Every detail, every angle is going to be important to a Ti user.
A Te user is much more capable of ignoring relatively unimportant angles and details. Something a Ti user could not possibly do, even though the value of said relatively unimportant angles and details may prove insignificant in what needs to be achieved. They still need to be considered.
Question (and this is not a loaded question, even though it may appear so - I am really trying to understand):
Following on @entropie's post, would you say that Ti is like he said, "Its more like puzzling information together leaving blank holes in between which fill over time and form the big picture in the end," whereas Te might be "taking the puzzle pieces and fitting them together"?
I'm trying to make a vague mental image for myself so I can understand....
You know what I believe? I believe a Ti user would never fully be able to explain how Ti works. That is just what Ti is.