Words mean things. People are using these two words quite a bit lately, and they are not interchangeable.
I just heard, for the umpteenth time, someone say "the government doesn't have the right to do this." That's true. The government has no rights at all. What the government has is the POWER to do things, and far too much of it at that.
Under our laws, only the people have rights, as in "the right of the people". Rights are a good thing. The government has nothing but power, and those powers are supposed to be limited to just powers exercised at the consent of the governed. Power is a dangerous thing, and should be strictly limited.
"Government is not reason, nor eloquence. It is force. Like fire, it is a dangerous servant and a fearful master" - George Washington
(This apt analogy explains that uneasy feeling you have that we're getting burned...)
Any addition to government power must by definition come at the expense of our rights. All increases in power are new paths to, and opportunities for corruption. It is becoming clearer and clearer that fear of government power is a healthy outlook. I would argue that it is the only possible outlook for those who wish to pass any semblance of liberty on to their kids and grandkids. (I would also argue that anyone who believes that unbridled government power can benefit society is mentally ill.)
In the context of Liberty, the word "rights" should bring a smile to our faces and the word "power" should strike fear. We used to understand these distinctions, and until we regain that understanding we cannot make our case effectively. Spread the word.