The word “art” floats in a cartoon cloud above the street in the architect’s sketchbook, on display in the new $305m (£209m) extension to the San Francisco Museum of Modern Art. “Environmental concept”, reads a scribbled note on the page. “Fog.”

Norwegian architects Snøhetta, designers of the chiselled 10-storey addition to SFMOMA, due to open to the public on 14 May, are fond of their natural metaphors. They talk of their spectacular opera house in Oslo as an iceberg, their cultural centre planned for the deserts of Saudi Arabia as a pile of pebbles, while here in San Francisco they have attempted to conjure something as light and vaporous as the city’s famous coastal mists.

Fog isn’t the first thing that springs to mind when you encounter the rippling white cliff face that now looms behind the museum’s original home, built in 1995 by Swiss po-mo maestro Mario Botta. It looks more like a gigantic meringue, a building-sized baked alaska slumped on the skyline between Botta’s weighty temple and the elegant Art Deco tower of the Pacific Bell building behind.

A view of SFMoMA at night: ‘A building-sized baked Alaska slumped on the skyline.’ Photograph: Iwan Baan/SFMoMA

“We wanted something much lighter, more open and transparent than our existing home,” says Neal Benezra, director of the museum since 2002. “Botta gave us this big, muscular, iconic presence that we needed when we first moved here, to what was then quite a run-down neighbourhood, but times have changed. Back then, a museum’s fundamental role was about taking care of and protecting the art, but this century it’s much more about the visitor experience.”

The vast new extension almost triples the museum’s display area to 175,000 square feet (16,250 square metres) – providing 40% more gallery space than even the Museum of Modern Art in New York – arranged as a stack of floors behind the existing building, wrapped in an amorphous shell that bulges out around its midriff, in order to fit more floorspace on the relatively narrow lot.

Clad in undulating panels of white fibre-reinforced polymer, which are intended to recall both fog and the rippling water in the bay, the building has a inescapable flimsiness, as if has been carved from polystyrene like the architectural models on display inside. The form is sliced flat where it meets the street, creating a shear surface “like a rock cracked open”, say the architects; but, clad with back-painted white glass, it instead presents the dumb elevation of a generic department store. Compared to the carefully detailed brickwork and banded granite blocks of the Botta building next door, the whole thing feels a bit cheap, in both material quality and architectural thinking.

A terrace walkway on the seventh floor. Photograph: Josh Edelson/AFP/Getty Images

The two buildings couldn’t be more different, or better reflections of their respective eras, and the pair do not make a particularly happy marriage. Indeed, after visiting the conjoined complex, it makes you wonder why one was deferentially kept at the expense of the other, while being considerably lobotomised in the process.

Botta’s stately museum squares up to Third Street with an imposing presence, standing as a stepped brick ziggurat from which a bold zebra-striped cylinder emerges, sliced at a sharp angle to form a striking cyclopean oculus. With a strong central axis of symmetry, the design sampled everything from Louis Kahn to Siena Cathedral, ushering gallery-goers in through a suitably momentous entrance sequence. At the bottom of the light-flooded cylindrical atrium, visitors were funnelled into an enclosed staircase of dark granite, rough and polished in alternating bands, a theatrical threshold that had the drama of entering a pharaoh’s pyramid.

All this has been swept away. In place of Botta’s cosmic portal now stands an open wooden stair, angled off-centre and rising in a jaunty dog-leg. The effect is like erecting an Ikea flat-pack in a temple, the blonde wood jarring with the weighty granite columns. The museum says that the original stair was too narrow to meet the projected increase in visitor numbers, and they wanted to open up a visual connection to the new foyer beyond – which, accessed by its own entrance stair from the other street, makes Botta’s lobby feel baggy and redundant.

The Pat and Bill Wilson Sculpture Terrace with an Alexander Calder sculpture. Photograph: Henrik Kam/Courtesy SFMoMA

The Ikea flatpack sensibility continues within the new building, where a world of more blonde maple and not particularly well-crafted details unfolds. It has the inoffensively Scandi style of a modern high-street clothing store (a Gap flagship perhaps), which might not come as such a surprise, given that primary purpose of the extension is to house the donation of the 1,000-strong collection of Doris and Donald Fisher – founders of Gap.

It is a vast amount of space, filled with 19 separate exhibitions for the opening season, which makes seeing the whole thing in one go quite an ordeal. It is a marathon journey that’s not particularly helped by the relentless uniformity of the galleries, none of which enjoy natural light or views to the outside world. Moments of respite are thankfully provided in the circulation area along the building’s northern flank, where progressively narrowing staircases rise between landings with deep window seats and views out to the city, while a couple of sculpture terraces also bring a bit of variety, one of them providing the dramatic feeling of edging along the side of a glacier.

But it is disappointing that the uppermost levels – visible from afar, where long windows cut across the facade, suggesting the promise of a panoramic viewing gallery or restaurant – are in fact the museum’s offices, off-limits to the public. It’s a boon for staff, who enjoy some of the most airy light-flooded offices in the city, with access to three outdoor terraces, but a strange decision not to make a public feature of this art mountain’s summit. It is particularly odd given that Benezra chose Snøhetta in the first place after being “mesmerised” by the mountainous, climbable landscape of their Oslo opera house, where the public is given free reign to clamber all over its sheer sloping stone roof. It is one of the best buildings of the last half century in his view – but one that maybe didn’t lend itself to being shoehorned into downtown San Francisco and grafted onto the back of his existing museum.