I'm going to put each of these noun phrases into the sentence "_ BE important to us" for sake of making them into full sentences. I'm also going to remove "on the internet and social media" in my examples, since that doesn't differ in meaning between the two of them.
Strictly speaking, these sentences both mean the same thing. However, the latter suffers from potential ambiguity. "Privacy and personal information protection are important to us" could mean either of the following:
Privacy is important to us, and personal information protection is important to us.
Privacy protection is important to us and personal information protection is important to us.
However, when you word it with "protection of" at the beginning, the two are easily distinguished by the verb form, since "protection of [privacy and personal information]" is singular but "[protection of privacy] and personal information" would be plural. So the former would be "Protection of privacy and personal information is important to us" whereas the latter would be "Protection of privacy and personal information are important to us."