The most distinctive thing about Luis Vidal's shiny new terminal is Richard Wilson's massive sculpture in the middle of it

The check-in hall at Luis Vidal's Heathrow Terminal 2: 'not good enough'. Photograph: LHR Airports Limited/Heathrow Photo Library

Imagine an airport as majestic as a cathedral, with the vitality and sociability of urban squares, and as enriching as a great art gallery. It would feel not like a piece of transport equipment, nor a shopping mall, but a piece of city. Usually airports are extensions of the space of international travel, and have more in common with the faraway terminals to which they connect than they do to the places where they are built. This dream airport would belong to the ground more than the air – the culture of its home city would begin at the arrivals gate.

Airports cover the same sort of area as, say, ancient Athens or Renaissance Florence, and their annual transient populations can be many times the number of inhabitants of modern London or New York. Why might they not, then, offer a fraction of these cities' art and enlightenment, their adventure, beauty and surprise? Why not have an airport that would be a destination in its own right, where people would want to go even when they had no plane to catch or meet?

Such a vision was offered last week at the launch of Heathrow's new £2.5bn Terminal 2, by its architect Luis Vidal and others involved in the commissioning of its art and architecture. Vidal said the new building is a London icon to match St Paul's and the Gherkin, and, pointing out that its central waiting area is the size of the covered market at Covent Garden, that it will be a "great social gathering space", like a piazza or a cathedral. Sir Peter Bazalgette, chair of Arts Council England, said that its Covered Court – the zone between the terminal and its car parks and transport connections – is the size of Tate Modern, and could offer an artistic experience to rival it.

Sadly, the new Terminal 2, which will handle 20 million of Heathrow's 70 million passengers per year, is not this wondrous airport. It is rather an iteration of what, since Stansted was built a quarter-century ago, has become the generic type – expansive oversailing roof, lots of steel and glass, big windows with views of tarmac and aeroplanes, floors in granite or terrazzo, lots of shops. As in Terminal 5 and many other places, the new building has two main levels, with departures placed above arrivals. This type is a significant improvement on the drab sheds that were often built before (see, for example, Heathrow's Terminal 4), but when Vidal stands on the balcony overlooking his would-be Covent Garden and announces that this is a wholly new type of airport, the only possible response is: no it's not.

The roof of the new Heathrow Terminal 2A. Photograph: Photo @ LHR Airports Limited/Heathrow Photo Library

Nor is it the finest of its type. There are some nice things about the new terminal, such as a generous supply of north light, some amplitude of space and clarity in the layout. Vidal promises functionality and well-adjusted acoustics, which can't be assessed fully until the terminal goes into operation in June but are worthy ambitions. I'd like to add its wavy vaults, clad in silicon-coated fabric, to this list of positives, but they are lumpily realised. Their curves grind and stutter, and suffer horrifying collisions where they meet the tops of glass walls or steel structure. They don't soar, they galumph.

At one point Foster + Partners were involved in the design of this building. As the architects of Stansted, Hong Kong and Beijing's pre-Olympic terminal, they are experts in this type of airport, and would have realised it more classily. Somehow they left the scene, for reasons that the man from the airport didn't explain. He did, however, say that he liked Vidal for "not being precious", "not clinging to his concept" and being "flexible".

I'm guessing that this is code for: "we didn't want a big-name architect who tells us what to do", and possibly also "we had enough of this with Richard Rogers at Terminal 5", which seems to have been a complex experience for many concerned. Maybe he just meant: "Heathrow's owners are still paying too much interest on their 2006 leveraged buyout to spend anything on design." In any event, flexibility is well and good, but not if it means accepting the clunks and crunches and uninvited bulkheads that mar Vidal's building. Terminal 5 has its compromises but it is still far more impressive.

Nor does the new terminal fulfil its makers' promise of capturing the spirit of London, except for embodying the British fondness for contractor-led construction that kills architectural ambitions. The retail offer will include a sprinkling of Anglo-quirkiness – Heston Blumenthal, Paul Smith, Cath Kidston – but otherwise there's little to tell you that you are not in Newark, or Düsseldorf.

It's not good enough for what, in the usual blather, is supposed to be a "world-class airport" serving "the greatest city in the world".

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Which leaves Slipstream, a sculpture in the Turbine Hall-sized Covered Court, to bear alone the task of uplifting passengers' experiences and letting them know that they are entering or leaving, as the blather also has it, the "cultural capital of the world". Weighing 77 tonnes, 78 metres long, and costing £2.5m to build, it might be big and strong enough for the task. Its artist, Richard Wilson, is also good enough, having previously created such compelling works as A Slice of Reality, a sawn-off ship next to the Millennium Dome, and 20:50, the walk-in tank of reflective oil once installed in the Saatchi Gallery.

Here he has some challenges. His work tends to flirt with disaster – a crunched-up aeroplane, buildings with holes in them, an upside-down house, the ship – but such associations would make nervous both airport authorities and passengers. A ripped-up Airbus, anyone? Airports require reassurance, and so Wilson goes for a more innocently upbeat idea, which is to make solid the shape carved in the air by a stunt plane, and realise it with riveted aluminium reminiscent of old-fashioned aeronautics.

It comes perilously close to being a large pointless thingy of a kind often installed in airports in the name of art, but there is enough force in its concept and execution to rescue it. It would be more powerful, however, if the space it occupied really were like the Turbine Hall; that is, if it had some resonance. As it is, it's like singing a cappella in Tesco, or drinking burgundy from a plastic cup: the container lets down the thing contained.

So the noble project of a city-like airport is left for another day and another place. Such an airport could learn from Grand Central Station, where the machinery of transport is out of sight, and the people are offered a great urban room to do with as they please. It might use brick, stone or timber, or adobe, mosaic or ceramic, and support itself with columns or solid walls, for where is it written that airports must always be of steel and glass and have wide-spanning vaults? It might have different qualities, as in cities, from one part to another.

There is a moment in Terminal 2 where you can glimpse this alternative. The open end of the hall containing Slipstream frames a rectangle of sky in which ascending aircraft appear, suddenly and unexpectedly close. There is drama in this encounter of aeronautics and architecture, and of the international space of flight with the urban space of the ground. It would be that much stronger if the building, with its metal and shine, were not trying to look like a machine.

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