One of the biggest issues with the original English dub was the insistence on changing dialogue, and also adding dialogue where there wasn't any before. The Japanese DBZ has a lot more silence than the English version, where they feel the need to fill every scene with either rock music or dialogue. You can see it even in the History of Trunks clip.
The dialogue is changed as:
00:34
Dub: Oh no... no...
Original: Gohan-San...
00:52
Dub: Gohan, why'd you have to do it? Why'd you have to leave me? It's not fair.
Original: Gohan-san... Gohan-san... Gohan-san...
01:12
Dub: Oh gosh, no... what'd they do to you Gohan?
Original: Gohan-san... (in the original there is silence after the camera pans down)
01:20
Dub: You were my best friend, you were everything to me... everything
Original: Gohan-san... (in the original there is silence until the shot of Trunks speaking)
01:32
Dub: This just isn't fair. Gohan, Gohan, Gohan!
Original: It can't be!
Now, in this instance I actually don't have a problem with what they did here. I think the English version is more emotional, but the 'emptiness' of the Japanese version and Trunks just repeatedly thinking/saying Gohan's name really emphasises how Trunks is in total shock. In the English version he's thinking coherently enough to have an inner monologue, but in the Japanese version he's so stunned that he can barely think or say anything other than Gohan's name. The tone of the scene between both is completely different, and this happens a lot in the original English dub. However, in this instance the music saved that scene. If it was generic synth/rock music it would have been far less impactful, although I think Trunks' English VA does a very good job in that scene and his scream at the end is actually better than the Japanese VA. It might be the best one in the entire English dub, actually.
Now, in a case where the dialogue is totally different and shit:
I won't go through it line-by-line, but none of this is in the original. No bizarre "cat loves food, yeah yeah yeah" song, Bulma isn't calling Yamcha a "goof" or a "nut" and she never mentions dreaming about Vegeta. When she's looking up at the sky the dub suggests she's thinking about Vegeta, but in the original, although she mentions Vegeta, she explicitly says that it's Goku she's worrying about.
So yeah, the English dub just took huge liberties with the scripts and sometimes changed them entirely. It got better during the Cell Games/Buu saga, but there's a lot of nonsense before that. The only positive is that when they released the DVD season sets they actually went back and re-dubbed a lot of the older material to make it more accurate, improve the VA performances and take out a lot of the stupidity. Not all of it, but a decent amount. The above scene in the re-dub is completely different and much closer to the original script.