A stock insurance company is structured like a “normal” company. It has shareholders (that are the company's investors), who elect a board of directors, who select the senior executive(s), who manage the people who run the actual company. The directors (and thus the executives and employees) have a legal responsibility to manage the company in a way which is beneficial for the shareholders, since the shareholders are the ultimate owner of the company.
A mutual insurance company is similar, except that the people holding policies are also the shareholders. That is, the policyholders are the ultimate owners of the company, and there generally aren't separate shareholders who are just “investing” in the company. These policyholder-shareholders elect the board of directors, who select the senior executive(s), who manage the people who run the actual company.
In practice, it probably doesn't really make a whole lot of difference, since even if you're just a "customer" and not an "owner" of the company, the company is still going to want to attract customers and act in a reasonable way toward them. Also, insurance companies are generally pretty heavily regulated in terms of what they can do, because governments really like them to remain solvent. It may be comforting to know that in a mutual insurance company the higher-ups are explicitly supposed to be working in your best interest, though, rather than in the interest of some random investors.
Some might object that being a shareholder may not give you a whole lot more rights than you had before. See, for example, this article from the Boston Globe, “At mutual insurance firms, big money for insiders but no say for ‘owners’ — policyholders”:
It has grown into something else entirely: an opaque, poorly understood, and often immensely profitable world in which some executives and insiders operate with minimal scrutiny and, no coincidence, often reap maximum personal rewards.
Policyholders, despite their status as owners, have no meaningful oversight of how mutual companies spend their money — whether to lower rates, pay dividends, or fund executive salaries and perks — and few avenues to challenge such decisions.
Another reason that one might not like the conversion is the specific details of how the current investor-shareholders are being paid back for their investment in the process of the conversion to mutual ownership, and what that might do to the funds on hand that are supposed to be there to keep the firm solvent for the policyholders. From another Boston Globe article on the conversion of SBLI to a mutual company, “Insurer SBLI wants to get banks out of its business,” professor Robert Wright is cautiously optimistic but wants to ensure the prior shareholders aren't overpaid:
Robert Wright, a professor in South Dakota who has studied insurance companies and owns an SBLI policy, said he would prefer the insurer to be a mutual company that doesn’t have to worry about the short-term needs of shareholders.
But he wants to ensure that SBLI doesn’t overpay the banks for their shares.
“It’s fine, as long as it’s a fair price,” he said.
That article also gives SBLI's president's statement as to why they think it's a good thing for policyholders:
If the banks remained shareholders, they would be likely to demand a greater share of the profits and eat into the dividends the insurance company currently pays to the 536,000 policyholders, about half of whom live in Massachusetts, said Jim Morgan, president of Woburn-based SBLI.
“We’re trying to protect the policyholders from having the dividends diluted,” Morgan said.
I'm not sure there's an obvious pros/cons list for either way, but I'd think that I'd prefer the mutual approach, just on the principle that the policyholders “ought” to be the owners, because the directors (and thus the executives and employees) are then legally required to manage the company in the best interest of the policyholders. I did cast a Yes vote in my proxy on whether SBLI ought to become a mutual company (I'm a SBLI term-life policyholder.) But policy terms aren't changing, and it'd be hard to tell for sure how it'd impact any dividends (I assume the whole-life policies must be the ones to pay dividends) or company solvency either way, since it's not like we'll get to run a scientific experiment trying it out both ways. I doubt you'd have a lot of regrets either way, whether it becomes a mutual company and you wish it hadn't or it doesn't become one and you wish it had.