Apparently both are measure words for flat objects, are they interchangeable or is there a difference to the types of flat objects they are used for?
张 literally means 'extend'. It is a classifier for something with a thin, flat surface , typically in rectangular shape
Typical objects with 张 as classifier:
一张纸 = a sheet of paper
一张刀 = a knife (refer to the broad side of the knife)
一张桌子 = a table (refer to the surface of the table)
一张照片 = a photograph (which has a flat surface)
面 literally means 'face'. It is a classifier for something with a 'face-side'. It can be in rectangular, oval, circular or other irregular shapes.
Typical objects with 面 as classifier:
一面镜子 = a mirror (only the face-side reflects image)
一面奖牌 = a medal plate (which has a face-side)
雷达 (radar) has a 'face side', but due to its huge size , 座 is used for its classifier, as in 一座雷达.
硬币 (coin) has a 'face side' but due to it's small size, 枚 is used for its classifier, as in 一枚硬币
张 (zhang1) is used for different objects, that may or may not be related with each other, e.g. sheets and flat furniture. If we really want to find a common denominator, I think that would be objects that are spread open :
- 纸 paper (一张纸 a sheet of paper)
- 地图 map
- 照片 photo, picture
- 桌子 table
- 床 bed
- 木板 plank
- 脸 face （一张脸）
- 嘴 mouth （一张嘴）
- 弓 (gong1) bow （一张弓）
面 (mian4) is used for flat objects with a face or a surface:
- 镜子 mirror
- 旗子 flag
- 墙、墙壁 wall
- 鼓 drum
- 锣 （luo2) gong
- 盾 (dun4) shield
 that's not a dictionary definition. It's personal advice that hopefully makes enough sense to you to work as a reasonable rule of thumb.
面 as a measure word is used for only a handful of objects, 墙 镜子 鼓 and 旗 are all I can come up with. It stresses that the object has a flat surface.
张 indicates that the object is flat and thin (usually can be held with two fingers, like paper or cards), while 面 doesn't.
Usually they are not interchangeable.