Does the palate expander make face wider?

It possibly could: Depending on how narrow the current palate is and how much expansion needs to be done. Typically, only slight, imperceptible facial changes occur. But other factors also play a role in how wide a face appears, including skin thickness, muscle tone, fat layer etc. If this treatment was recommended, it is to allow improved space to better allow your teeth to orthodontically reposition.

Not noticeably : A palate expander spreads the midline of the upper jawbone less than a centimeter. Since the appliance works symmetrically, half the total spread adds a minuscule amount of width to the bone to create a few mm of space in the arch for the teeth. The facial width is determined primarily by the cheekbones, muscles and fat deposits under the skin.

Smile wider...: Palate expander will expand the intermolar distance so that it would definitely increase the width of your smile, but not so noticeably in the facial width. It also will lower the palatal vault and increase nasal volume and thus help to treat oxygen deficiency which in many case eliminates the black ring beneath your eyes. This would change the appearance of your face.

Age dependent: In growing children widening the palate gives additional breadth to the upper jaw, resulting in appearance of higher cheek bones. In older teens and adults, after upper jaw bones have fused together, expander can only tip teeth outward, not expand jaw, unless the 2 halves of the upper jaw are surgically released.

It can: orthodontic treatment, as a whole, can make subtle improvements to the overall shape of your face. The operative word here is improvement. If you have a proper occlusion, your face will have a more balance and aesthetically pleasing look.