At issue is the fact that the FDA defines mayonnaise as containing egg yolk, and Just Mayo is a vegan alternative that uses yellow pea instead of egg. Unilever claims that Hampton Creek is falsely advertising their product as mayonnaise (even though it only uses the term "mayo"), as the word is defined by the FDA. By the way, you now have a hint as to why Miracle Whip isn't labeled as mayonnaise.
Most of what I've read on the web has been harshly critical of Unilever. This piece in Mother Jones is one example, framing the issue as a David vs. Goliath (or "Big Mayo"), and an NPR piece is titled "Bit Mayo vs. Little Mayo".
I've been interviewed by a couple reporters about the suit (e.g., see this CBC story), and I have to admit to having mixed feelings about the case. I have found it a bit surprising that many of the folks I normally think about as being in favor of consumer protection have been sided with Hampton Creek, I suspect because it is more appealing to take the position against any food giant.
What are the issues?
Going back to some of the concerns over 100 years ago that led to the passage of the Pure Food Act, there have been attempts to regulate the usage of certain terms and labels. The idea is that if you see a food labeled "mayonnaise", for example, that you can be sure it hasn't been adulterated - that food processors haven't watered down the product or used sawdust or some such nefarious substance to reduce cost. These sorts of rules exist for all kinds of food products (here, for example, is a list of approved labeling terms for meat from the USDA).
From Unilever's perspective, they're just following the FDA labeling rules that have ostensibly been created to protect the consumer against fraudulent labeling. Thus, I can see their point of view that Just Mayo is "unfairly" competing by implying they're making a product that does not conform with the FDA guidelines. While this may seem like a silly case, the implications could be much wider; when should a firm be able to use labels with the term "organic" or "free range" or "milk" or "bread"? Should certain standards be met to use these terms so that consumers aren't mislead?
On the other side, these sorts of rules could very well prohibit innovation and competition. I haven't tasted Just Mayo, but supposing it is comparable to "regular" mayonnaise, they've created a process that removes an animal ingredient, which is appealing to a segment of consumers. Moreover, what if a company can create a less expensive, non-egg based mayonnaise that consumers find tasty? Wouldn't the upstart have a hard time entering the marketplace and competing if they are unable to describe their product using word that best fits it? A lot of the labeling rules seem to be just plain bureaucratic overreach, and they seem to do more to line the pockets of lawyers and regulators than protect the consumer.
Ultimately, I have a hard time seeing how these kinds of name or labeling rules make much sense for products like mayonnaise. These are branded products. If a consumer buys Just Mayo and doesn't like the taste as well as conventional mayonnaise, they can go back to their old brand. In econ-speak, these are experienced goods, and firms have reputations that allow consumers to easily search and identify eating satisfaction with different brands. There seems little reason the market couldn't work this out.
Ultimately, I think there are good arguments on both sides of this case, and it isn't obvious what would be the consequences of the unraveling of these sorts of "food purity" laws. Sometimes it's hard to know when consumer protection becomes protectionism.