When Glenn Lowry, director of the Museum of Modern Art in New York, was welcomed into Tate Modern recently by Nicholas Serota, it was a meeting of arguably the two most influential museum directors in the world.
Lowry was in London to promote MoMA's forthcoming shows - including the first full retrospective of the South African-born painter Marlene Dumas, and a major show on Martin Kippenberger. In Tate Modern, all was courteousness as Serota, the impeccably besuited director of the Tate, introduced the equally trim and stylish Lowry to his British audience. Lowry extravagantly praised the Cy Twombly exhibition, curated by Serota, that has just opened at Tate Modern, calling it "a model of what we all seek to achieve" and "a truly wonderful show".
Seemingly, however, there was a subtler agenda in view for the American, a former competitive athlete - the lingering sense that part of his intention was to assert the authority of MoMA over that young upstart, Tate Modern.
In one sense, any competition between the institutions, and their powerful directors, is like that between David and Goliath. In terms of its collection, as Serota readily acknowledges, MoMA wins hands down, its extraordinary lineup of modern masterpieces effortlessly outclassing Tate's.
But arguably the Tate brand, which has gained huge international currency since the founding of Tate Modern at the turn of the millennium, has threatened to eclipse that of MoMA. The Tate has even been making audacious inroads into MoMA's home turf of New York, courting American philanthropists and holding glamorous fundraising events. According to Serota, MoMA has actually asked its trustees "not to join high-profile committees to help fundraise for Tate" - a claim that Lowry denies.
Lowry's take on the Tate's flourishing is simple. "Buildings come and go, directors come and go. The art is the constant," he said - alluding, no doubt, to the current fashionability of Tate Modern's gargantuan Bankside home no less than to MoMA's big-names collection.
However, the institutions' identities go deeper even than their collections, their buildings and their directors - they encompass the very principles on which the museums were founded. MoMA was created as a privately run and funded organisation; the Tate has always been a publicly accountable national institution that forms part of British civic life.
"The Tate is an instrument of state in a way that MoMA could never be. For good - and for bad," says Lowry. Needless to say, that claim is denied by Serota. He puts the difference another way: "We are an instrument of the public - which owns the Tate - while MoMA is governed by a very small group of wealthy individuals, and so it necessarily reflects the values of a narrower cross-section of society."
Museums, on one level, are just boxes containing pictures or artefacts. So do these questions of principle really, in the end, matter? One answer is that numbers speak for themselves. In the US fiscal year 2007-08, 2.7 million people visited MoMA (full-price entry, $20). Visitors to Tate Modern, by contrast, numbered 5.2 million (free).
Earlier this year Neil MacGregor, the hugely successful director of the British Museum, was courted to become the next boss of the Metropolitan Museum of Art in New York. He declined to be a candidate, he has said, because "the Met is not a public museum - whereas the British Museum is a public institution and the public museums of London have always been free to everyone".
Last year the British Museum had a record 6 million visitors, including 35,000 on one day to celebrate the Chinese new year. For the first time since the Chartist riots of 1848, its gates on Great Russell Street had to be shut to prevent more people coming in. We have good reason, then, to be proud of our national museums. Both the Tate and the British Museum (and many other arts institutions) have pulled off a terrific coup - they have a meaningful and deep relationship with the varied and rich populations that surround them, as well as a profile in the wider world.
In practical terms it is time, then, to support both the Tate and the British Museum as they embark on ambitious growth projects. The museum is planning to build new galleries, costing £100m, to give its increasingly strong shows much more room to breathe; while the Tate wants to spend £250m on an extension that will increase its space by 60%. Both these schemes are aimed at a 2012 completion, in time for the London Olympics. In the current downturn both may, without immediate help, struggle to raise the vast amounts needed. They must not be allowed to miss their deadlines.
More importantly, perhaps, we should revel in our good fortune, as members of the public, that we are the owners of these institutions; we should value them, love them, and argue with them. And, in the occasional moment of complacency, we might even allow ourselves to recall who won when David and Goliath fought.