I have wondered about this myself (believe it or not) and come to the conclusion that this is to help minimize the amount of cleaning necessary. Without getting too graphic, the front section of the toilet seat is the section most likely to have repeated dirty liquids on it. This is particularly the case in a men's room, as toilets will often be used instead of urinals at busy times. It may not apply as much in the women's room, but not having the front section does not in any way affect the use of the seat, so it is easiest to have one type of seat for all toilets in a building. Should men using a toilet standing simply lift the seat: yes. But (a) the seat often doesn't stay up well and (b) many people don't want to touch the seat at all, for the same reason that disposable seat covers are supplied in many public bathrooms.
The lid is a combination of cleaning - if you don't have it, you don't have to clean it - and aesthetics. Simply put, the lid in a home bathroom keeps things relatively out-of-sight when using the bath or shower or sink. But in a public bathroom, most people spend as little time as possible in the bathroom (leaving more extensive grooming for their personal bathroom at home) and the toilets, except in a single-toilet bathroom, are behind a door so they are out-of-sight when using the sink anyway. A lid is also useful to sit on a toilet as a chair, which could be useful occasionally in a public toilet (e.g., if you are using a stall to change clothing), but is definitely useful at home.