The source is Bostrom's 2004 paper A history of transhumanist thought, page 4. I'll paraphrase the difference he lists:

Transhumanism uses tech to change bodies and minds, Nietzsche uses old pathways.

Yeah, that's his mistake. He points at the right goal, but can't say how to get there. As I said, no real work.

Transhumanism wants to boost everyone, Nietzsche only a select few.

I think that's unfair to Freddy. His Zarathustra puppet goes around telling everyone to do it, but they aren't interested. Obviously he was envisioning individual progress as opposed to inventing tech then distributing it to Muggles, so he thinks that if few people want to put in the effort then few people will get boosted.

Transhumanism likes individual liberties.

I don't understand what Bostrom means by that. AFAICT, Fred is huge on individual liberties.

Transhumanism comes from the Enlightenment.

I fail to see the relevance.

What I got from reading Nietzsche (before I got any exposure to transhumanism) was an extremely pretty way of saying "Striving to improve yourself a lot is awesome". No argument why, no proposed methods, some very sucky assumptions about what it'd be like. Just a cheer, and an invitation for people who share this goal to band together and work on it. Which is what transhumanists have done.