There are essentially two points of debate. One is whether the conservation of whales demands we maintain the near-moratorium (in addition to the scientific clause, some whaling by aboriginal peoples in coastal waters is allowed, including in Alaska.) (Smaller whales and dolphins, BTW, are not covered and are still subject to major hunts in Japan, Norway, and elsewhere.) The other major question is whether it is ever morally acceptable to harvest some of the world's most intelligent mammals, given that, while they are certainly useful to aboriginal communities, no human population absolutely needs whales to survive.
The minke whale, subject of most current whale hunting (NOAA)
At the very least, we need to maintain the moratorium and close the "scientific" loophole. Japanese whalers who claim they take only minke whales (the second-smallest and by far the most numerous of the baleen whales) under the science clause have in fact taken other species, as meat in Japanese markets has been proven by DNA sampling to come from humpbacks and other species. (In one case, there was no question because a sample found in a market in 1993 could be identified as a specific whale: an incredibly rare blue/fin hybrid, killed off Iceland in 1989.) That can't happen by mistake. It can only happen by deliberate negligence. Japan argues much legitimate science has come out of its program, including the astonishing discovery of Omura's whale, a 10m long animal described only in 2003. Japan's opponents argue the science hasn't been nearly worth the cost to get it.
I argue that, even if the science exemption was valid, Japan has forfeited the right to use it by taking species they say they are not taking. Japan argues they are being singled out, since Norway takes several hundred minkes under a formal objection to the moratorium (an artifact of the Vienna Convention which governs treaties: to summarize, you can lodge a formal objection and do something prohibited, but the other signatories have the right, individually or collectively, to impose sanctions and otherwise pressure you.)
But the fact Norway has been treated a little easier than Japan doesn't make Japan right. It just makes Norway also wrong. Protecting the large baleen whales still in great danger (blue, fin, etc.) requires that whale be kept off the market. That does mean sacrificing any science coming out of hunting, but as non-invasive approaches including the use of UAVs keep becoming more capable, it's an acceptable trade. Anything other than continuing the moratorium keeps up the pressure on the huge, smart, and emotional animals we nearly wiped off the planet.