A Community Interest Company is a type of company. It is not a legal form in its own right. A company that isn't a CIC can register to become one, but not vice-versa.
CIC status entails that the company operates in a particular way:
There are three type of CIC: one for companies limited by guarantee ("Schedule 1"), and two for companies limited by shares ("Schedule 2" and "Schedule 3"). Schedule 1 and Schedule 2 CICs operate similarly: they basically can't distribute profits. Unusually, this makes companies limited by shares and companies limited by guarantee, if they are CICs, operate in a fairly similar way; normally the, the incentive of dividends leads to very different behaviours by those two types of company. Schedule 3 is more like a normal company that isn't a CIC: it can pay dividends, up to a point.
Now, you could write a company's articles of association to achieve the same thing without registering as a CIC and dealing with the extra faff. However, since 2006 there has been nothing to stop the members/sharesholders of a company from overriding the company's articles by voting in new ones; entrenched provisions of the articles may be amended by unanimous vote. If this were allowed in CICs, then they could solicit donations and contributions on the basis of being a CIC, and then amend their articles to allow the money to be spent on parties and alcohol for the shareholders.
But CICs are not allowed to remove the restrictions from their articles. These restrictions are basically a set of prescribed words which must be in the articles, and a CIC must choose between the three schedules of the relevant regulations, and adopt one of the three forms discussed above. The system of prescribed articles is a good one, and is used for Right-To-Manage companies taking over blocks of flats too.
So what's the problem?
Well, for CICs, there isn't a problem. It's just that for every other type of company, constitutional protections like this are no longer available.