There are no character-level differences. Hiragana and katakana are, for all intents and purposes, the same, differing only in how they are used with regard to the broader idea of choice of system. You say you know what each is used for, so that's the key distinction you need to focus on.
I think one thing we might be able to mention is elongated vowels. Where in hiragana you would write ああ・いい・うう・えい・おう in katakana you would usually write アー・イー・ウー・エー・オー respectively. Note though that for certain effects or spelling sounds that aren't in words, like a scream or a sigh or something, you may very well see ー used with hiragana.
Another aspect that we might be able to say fits the question's criteria is that katakana is often used to represent foreign pronunciation or otherwise weird ways of talking. There isn't a standard, though, and it's more the choice of katakana in itself that acts as a kind of visual metaphor for different pronunciation. The characters themselves would not indicate any specific change in pronunciation, though, so I don't think we can say this is an example of katakana being pronounced differently as much as it is the form that suggests it, if that makes sense. Maybe compare it to some angular robot-looking font in English being used for robot speech. You wouldn't look at
a word written for robots and say it's pronounced differently.
Some characters in katakana simply aren't seen very much because of how katakana is usually used. For example, grammatical bits are usually written in hiragana, so it's pretty rare to see ヲ in katakana (since を is nearly always used as a particle). But still you do see it from time to time and it's not worth writing it off either.
Similarly, small kana that's used to approximate foreign sounds that aren't in Japanese usually aren't written in hiragana, so you don't see that much unless it's intentional. For example, ふぁ ふぃ ふぇ, etc. In most real applications these sounds/kana combinations would be in katakana. As is the pattern, though, you will see both.
Katakana and hiragana are a one-to-one mapping. Like different fonts, basically: they're the same, but they're used in different situations. Most distinctions that you make will fall into the "usually but not always" category anyway. Focus on the more meta idea of what the choice of system in itself means and not so much on specific character level differences.