Unfortunately, you can't point to a single event, or even series of events and say: "That is Climate Change". It's far more generalized and gradual than a single event. There is no need for panic in the streets, not that you could run away from it really.
Usually when you talk about Climate Change, most people talk in terms of the misnomer: Global warming (see the excerpt from Futurama). The term Global Warming doesn't really describe the totality of the phenomena we are seeing today. Sure, the Earth is heating up at a much faster rate than we have seen in known history, but it's more than just added heat. The global scale of the environmental impact our society has is truly astounding. From deforestation around the globe to acidification which is destroying coral reefs at an alarming rate. Our planet is chemically, biologically, and physically sick.
Right now we see the symptoms of the illness: warmer weather, more severe storms, more frequent storms, widespread drought, depletion of aquifers, changing weather patterns, changing crop yields, receding glaciers, increased coastal erosion, and rising sea levels. The difficulty is altering the perception in skeptical minds. There are arguments on both sides leading to confrontations like this:
No matter how correct Professor Cox may be, confrontation won't win a debate in the court of public opinion. I wasn't able to find the full debate, but from what the BBC posted both parties had equally valid points. Mr. Roberts' contention was that he believed the data was manipulated and faulty. Professor Cox's response was simply to chuck the data at him, not address the point of validity. If the scientific community really wishes to address the issue properly, they have to address the issue of credibility. That's a bit difficult when people point to problems with the system:
Some of this has to do with the idea of mistaking correlation for causation, but a lot of it has to do with the funding the science receives. If the entity funding the research or the researcher has an agenda, then the data has a significant chance of being skewed in favor the the agenda. If any tampering exists, or is even perceived to exist, then it is nearly impossible to overcome the skepticism which should accompany any scientific research. This is where Professor Cox runs into problems. He believes the data is correct, but cannot provide sufficient proof to overcome the burden of proof required to sway others.
It is true that Mr. Roberts will probably never have sufficient evidence to convince him that Climate Change is real, but that stubbornness can't be debated away. He has to be receptive to the idea that his null hypothesis is wrong. Without an attitude change, there won't be any possibility of any meaningful action.
This isn't an issue that is going away. This is something that will only get worse. While it slowly happens over the next few decades and centuries, it will cost staggering amounts of money and take a great toll on life. They won't be dollars spent on a single disaster or lives lost in a moment. It will be a slow and steady churn that will be ascribed to a storm like Katrina, or Winston.
Of course, we could always have a bare-knuckles cage match between Mr. Roberts and Professor Cox. Somehow I don't think science would win with the skinny particle physicist taking on the lumbering conservative.