Bratislava has the distinction of having been the capital of two different countries in its time. For several hundred years it was the capital of Hungary while Pest was occupied by the Turks. Now, of course, it is the capital of Slovakia.
For furthest one I’d wag Sydney, or whatever the capitol of Greenland is, or perhaps some other mid ocean island.
Ryanair make the most of this, too, using their infamous tactic of geography based on fisherman’s estimates. Cheap flights to ‘Bratislava (Vienna)’.
What about Bahrain-Qatar? Manama-Doha looks to be about 60 miles, getting close to the European competition.
Finding the distance between two points is pretty easy with this site. Using it, the distance from Canberra to Port Moresby to be 1784 miles.
Kigali (capital of Rwanda) and Bujumbura (capital of Burundi) are only about 110 miles apart.
According to the Maporama website, Vienna to Bratislava is 64 km (40 miles) city-centre to city-centre.
Home of Peter “Lord of the Rings” Jackson?
Brazzaville / Kinshasa is 6 miles
For most distant capitals of countries that border each other Moscow / Bejing is 3610 mi (5810km)
Brian
Greenland belongs to Denmark, although it’s semi-autonomous. Does it count? Besides, Reykjavik is too close to have Godthab in the running (and Reykjavik, in turn, is too close to London - about 1200 mi).
Using the site supplied by MikeS:
The closest capital to Canberra is Wellington – 2327 km / 1446 miles.
I think that’s the one to beat for greatest distance.
According to Wikipedia, both Brazzaville and Kinshasa are at 4°16′S, 15°17′E, which would mean they have zero distance between them. However, looking at Google Maps, the centre of Kinshasa is closer to 4°20′S, 15°20′E, which would give a distance of 8 km or 5 miles between city centres.
And if we’re going to count semi-autonomous places, Tristan de Cunha is probably a strong contender.
No, no, in an Italy shop. The Vatican is nowhere near China. 
And while we’re at it, Bratislava was known mostly by its German name, Pressburg, at that time. To Hungarians, it was known (and still is) as Pozsony.
Yes, the city only acquired its present name after WW1
Helsinki (Finland) to Tallinn (Estonia) is only a 50-mile ferry ride across the Gulf of Finland. Nowhere near as close as Kinshasa / Brazzaville, but getting down there with Vienna / Bratislava.
How close is Monaco to nearby European capitals (either Paris or Geneva, maybe)? That might be a close one.
You mispelled Bern.
I’m not sure if Vatican, Monaco, San Marino, etc. really have capitals.
There was a joke “contest” here once, really just a freebie giveaway promotion, in which the only question asked was “What is the capital of Singapore?” Some countries are just their own capital I guess; should that count?
What’s the closest capital to Papeete? (the capital of French Polynesia, just on the remote offchance any dopers didn’t know that)
Most of the micro-countries actually do cover a reasonable amount of land, perhaps equivalent to a U.S. or English county (which is about as variable an area as they are).
Luxembourg, for example, has two distinct areas of terrain, and a fairly strong industrial sector in its economy – as opposed to the other nations which survive on being tax havens, selling postage stamps, etc., Luxembourg is big enough to have a “normal” economy, albeit a relatively small one.
There are a number of other towns in Luxembourg, Lichtenstein, San Marino, etc. IIRC, Vaduz is not even the largest city in Lichtenstein – though the community that is, is familiar only to Swiss and Austrians where it’s a major stop on the international rail service. Even Monaco, small though it is, has four “urban communes” (i.e., cities), though their borders abut each other.
San Marino the capital city is, again IIRC, the only community of size in the country San Marino – but there are a number of rural crossroads villages of some local importance.
I believe Singapore the city has sprawled across the whole island now, but during WWII there were jungles and villages along the strait in the north separating it from the mainland, while the city proper lay on the southern coast.
Most of the Polynesian states include either a fair amount of land area with scattered settlements, or a variety of atolls separated by lots of water. Even Nauru and Niue, the two (one fully independent, the other an autonomous state in free association with New Zealand), the only two single-island nations in the South Pacific, have a capital and a bunch of outlying villages/settlements.
Nuku’alofa, Tonga and Apia, Samoa both appear to be closer than the distance between Wellington and Canberra. Anyway, if we’re allowing possessions like French Polynesia, Pago Pago in American Samoa is even closer, and Avarua in the Cook Islands (NZ) is only about half the distance of Pago Pago.
Not independent. The capital of the nation which includes French Polynesia is named Paris. 
Which reminds me of a no-longer-quite-true trivia question: Where, if anywhere, is there a land boundary between France and the Netherlands? (There was, and it was on the West Indian island of St. Martin. However, the Netherlander half of the island is now part of the autonomous Kingdom of the Netherlands Antilles.)
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I believe Singapore the city has sprawled across the whole island now, but during WWII there were jungles and villages along the strait in the north separating it from the mainland, while the city proper lay on the southern coast.
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You can still find some rural areas in the country, especially in the northern part of the island. But the whole island seems to be considered one entity from what I can tell, rather than, say, the city versus the rest of the island. Singapore also includes a few other small islands that tend to be forgotten.