It would be very useful to know whether the rose in Monica's photo sets hips and additional close-up photos of leaves, buds and inflorescences would be helpful.
This conversation sent me out into the garden to collect bits and pieces from 'Marie Van Houtte', 'Climbing White Maman Cochet' and 'Maman Cochet' to scan.
Have done some comparative scans and will post them here.
It was not possible to compare the blooms or buds of 'Marie Van Houtte' and 'Climbing White Maman Cochet' because, while 'Marie Van Houtte' is growing actively at the moment, putting out new red flowering growth, 'Clg White Maman Cochet' is having a rest. It is covered with healthy foliage but is not producing new flowering growth at the moment.
'Maman Cochet' is also slowing down for winter but has one inflorescence, so I've used this to compare with that of 'Marie Van Houtte'.
Prickles: Both 'Marie Van Houtte' and 'Climbing White Maman Cochet' are very sharp to handle, with well armed canes, laterals and leaves.
Hips?: If the spent blooms are not removed from 'Marie Van Houtte', it tends to form many hips that contain seeds. It is very rare for 'Clg White Maman Cochet' to form hips. Most uncut blooms simply shrivel.
Pedicels: The pedicels and receptacles of 'Clg White Maman Cochet' (and other Cochet forms and colours) are, in my experience, consistently smooth.
'Marie Van Houtte', on the other hand has pedicels that vary in their glandularity. They may be completely smooth, well covered with stalked glands, scattered lightly with glands, or have pedicels that are smooth on one side and glandular on the other. Examining an inflorescence of three buds today, the central bud had a smooth pedicel while the two side buds had glandular pedicels and this is a pattern that is seen quite often on 'Marie van Houtte'. On another cane, where the three spent blooms were developing into hips, the pedicel of the central hip was smooth while the outer two felt like sandpaper.
Foliage: The leaves of 'Marie Van Houtte' and 'Climbing White Maman Cochet' differ in colour and, to some extent, in size and shape and in the depth of their impressed veins.
The foliage of 'Marie Van Houtte' appears darker, somewhat narrower and while the veins are impressed, they are not as deeply impressed as those of the Cochets.
The foliage of 'Clg White Maman Cochet' - like the shrub form and like 'Maman Cochet', is a lighter green and it tends to look brighter and to accentuate the impressed veins, giving the leaves a shiny 'quilted' appearance.
Taking into account that Tea rose leaves tend to broaden slightly as they age, and that most Teas will have leaves with a range of shapes and sizes, overall, the foliage of 'Marie Van Houtte' is narrower than that of 'Clg White Maman Cochet'.
Colour of new growth: The new growth of 'Marie Van Houtte' is a rich red - tending to a purplish red while that of 'Maman Cochet' and 'White Maman Cochet' tends to a brownish red.
Bloom colour: As a general observation, both 'Marie Van Houtte' and 'Climbing White Maman Cochet' can produce flowers with a base colour that is very pale - almost white, but while the base colour of 'White Maman Cochet' tends to range from almost white to pale, creamy yellow, those of 'Marie Van Houtte' can get much deeper, particularly in cooler weather, when the blooms would be described as a true yellow.
Buds: The buds of 'Marie Van Houtte' are plump and, generally shorter than those of the Cochets.
Inflorescences: The pedicels of the Cochet roses (by this I mean 'Maman Cochet', 'White Maman Cochet' and their respective climbing forms) tend to be longer, the inflorescences larger, and all things being equal, the blooms of the Cochets tend to be larger.

I'll post this and then try to add some images.