The first thing you need to bear in mind is that we're toalking about US definitions here, NOT UK ones. While there are many similarities, there are also differences.
A US Corporation is more of less the same as a UK Limited Company, in that there is separation of ownership from control (i.e. shareholder and directors), limited liability on the part of shareholders (generally, at least), and taxation based on company law.
An LLC (in the US) is a form of business structure that combines many of the features of a company (like limited liability) with the features of a partnership, but has "members", not shareholders or partners. It's a kind of hybrid. But there are considerable implications in terms of the legal structure, reporting requirements, and in terms of profit share and hence, tax implications.
Any incorporated body pays tax on profits at corporate rates, then passes some or all of the excess to owners as dividends, bonuses etc. The owners only pay personal taxes on distributed profits. So, if you retain alarge proportion of funds inside a company to fund growth, personal tax liability is reduced and you have a tax benefit that favours capital growth over income generation. Unincorporated bodies pass all tax liability for profits to the owners, even where funds are retained in the business (to fund expansion, perhaps), and owners therefore pay tax at personal tax rates.
And so on. The differences are numerous, and some of them are quite subtle.