This is an interesting question, partly due to the geographical nature of what people assume empires should be, versus what empires actually were historically.

In classical history empires were defined by geographic areas that were large enough to encompass more than one 'kingdom', such as the Roman or Greek empires, such that when those empires collapsed the constituent areas fell back to being duchies or kingdoms.

In the Victorian and Napoleonic sense the term 'empire' was applied to both countries that had colonial possessions, such as France and Britain, and self-declared empires such as Abyssinia (which has conquered the neighbouring kingdoms). While the British came to be seen as an imperial nation, indeed were known colloquially as the 'British Empire', the actual name of Britain remained the United Kingdom of Great Britain and Northern Ireland. Additionally, when the United States gained colonies the US government went out of its way to avoid being seen as imperial, rather they sought to be seen as a civilising influence on Cuba, the Philippines, and Hawaii et al (which carries over into present day colonial polices with Puerto Rico).

France is an interesting example, as under the pre-revolutionary Bourbon monarchs France was a Kingdom, then when Napoleon became ruler for life he rejected the title of King in favour of Emperor of the Republic, which in turn created the first French Empire. Upon restoration of the Bourbon monarchy France reverted to a Kingdom, and then when Louis Napoleon became monarch in 1852 he declared a second French Empire, which included colonial territories in northern Africa, India, and indo-China. In 1871 France once more became a republic, yet retained its colonial possessions but without being labelled an Empire republic.

An example of an anachronistic empire existed during the late medieval/early modern period under the Holy Roman Empire (HRE), which essential became a titular title as the Germanies fragmented into smaller and smaller constituent parts, with the Emperor only retaining authority with the tacit approval of the electors and states under their nominal demesne. When Napoleon formally abolished the Empire in 1806 is was a political gesture rather than an actual dissolution of an empire.

It is arguable that the difference between a Kingdom and an Empire is semantic, the personal preference of the monarch or nation proclaiming said empire. Many empires have existed without declaring themselves such, and often they are viewed as empires because of both territorial possessions and political will, such as the USA or the British Empire. Indeed, the concept of empire has evolved significantly since the Roman empire, to the point that where a country such as the USA or modern China could be viewed as historical empires due to its size and population density, they prefer to see themselves as republican nation states without imperial aspirations.

So, to answer your question, I would argue that an empire is a semantic term applied by nations that wish to assert their imperial aspirations, and that not every empire sees itself as such. The difference between a kingdom and an empire is one of declaration, external diplomatic recognition of that that declaration, and historical recording of said empire.