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What Jung has to say about the differences:
But just as little as it is given to extraverted thinking to wrest a really sound inductive idea from concrete facts or ever to create new ones, does it lie in the power of introverted thinking to translate its original image into an idea adequately adapted to the facts. For, as in the former case the purely empirical heaping together of facts paralyses thought and smothers their meaning, so in the latter case introverted thinking shows a dangerous tendency to coerce facts into the shape of its image, or by ignoring them altogether, to unfold its phantasy image in freedom.
Generally Te is interested in the goal and what logical steps to take. It does not check it's thoughts for logical consistency because the evidence, or what it can externally measure, is either true or false. It will use external facts to inform it's thinking and make judgments.

For Ti, the thoughts must be logical regardless of what externally is true/false. A thing is true/false because the reasoning is correct (and I think this is where Ti principles come into it, finding through logic universal truths that are always true or always false). Ti will come up with it's own ideas or theories through it's own logic and may (or may not) fit external facts to it's ideas.

Also an analogy I came up with an analogy a while back. It isn't perfect but I think highlights the differences, Ti, Te and cake:

Both Te and Ti are given a recipie to make a cake.
So Te starts by making a plan (Looks up a good five star recipe online, what in ingredients are needed, where to get them from, what tools will be needed, etc.)
Then will get straight on to prep work, making it, bake it, then let it cool down. Fine tuning the process to get the cake completed in the least amount of time.

In the mean time Ti looked at the recipie and asked why? Why these ingredients? Why in this order? Why these ratios. And then Ti begins the long process of experimenting with different ratios, cooking times, even ingredients. Eventually (with a few disasters along the way, like cooking the cake until it's a pile of hot ash haha) Ti finds the exact formula to make the best cake, with parameters that can be adjusted depending on the conditions the cake is to be made.

At this point Te looks at Ti and says "big woop, it's only cake" and is happily sitting down eating it's 20th cake compared to the one Ti has come up with in the same amount if time.
Ti looked at it's cake and thinks...
this is the one cake...the one cake to rule them all !!! *insert evil laugh here*
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