It's because of the current treaties (emphasis mine):
The Treaty on the European Union states that any European country may apply for membership if it respects the democratic values of the EU and is committed to promoting them.
To the best of my knowledge the European criteria has always been around. There have been debate over what the criteria is to qualify as a European country. Examples off the top of my head:
Some put cultural criteria forward. Religious ones in particular (i.e. must have Christian roots), so as to deny entry to Turkey. But then, consider Bosnia and Herzegovina, which is well on its way to joining, or Albania. There's little if any contention on letting the latter two countries join the EU.
I'm not aware of any similar arguments having been made against orthodox countries when members all had Catholic or Protestant roots. The point would be moot today, since Greece joined, then Cyprus and Romania, and soon Serbia. Why not throw in Moldavia, Ukraine, and Belarus.
The other main criteria has been strictly geographical. Turkey and Russia, the argument goes, should be allowed in because they've territory in Europe; or denied entry, because they've vast swaths of territory in Asia. Non-contentious countries, like Iceland and Norway, would of course be allowed in.
As an aside, Georgia should be allowed to join based on geography, since it's part of Europe. And sure enough, it's in talks about doing so. As a thought experiment, consider whether Armenia and Azerbaijan, both part of the Caucasus and the EaP, should eventually be allowed in too. (Yes, that would mean the EU having a border with Iran if either did.)
Yet another criteria combines a bit of both aspects, and revolves around the Mediterranean basin in the sense that the entire area shares ties owing to the Roman empire. It has been around since the early days of the EU - and even had proponents in the UK, of all places. (As a thought experiment, why not also consider Hellenistic heritage if that argument ever prevails. That would take the EU as far east as Pakistan and Afghanistan.)
At any rate, your specific questions:
But why limit membership to European countries?
What a European country is isn't so clearcut, and subject to a lot of debate in practice.
What difference does it make to existing EU members?
Many things, because citizens in member states don't all agree to what is admissible. In particular a great many Europeans aren't too keen on having free movement between e.g. North African countries and Continental Europe. Plus, you need to keep protecting borders in mind.
If a country is sufficiently developed and willing to adhere to EU rules, why not admit it into the Union?
It's already being explored, in a sense. See for instance Turkey - which many in Europe would argue is not European, even though the Ottoman Empire traditionally has been viewed as a European power and a successor state of the Roman Empire.
