As far as I know, there is no big difference between philharmonic and symphony Orchestra. Both types of orchestras are the same in so many areas. Orchestra simply means a large group of musicians who come together to play on various instruments, which usually includes instruments like brass, percussion, woodwind, and strings.
The little difference between philharmonic and symphony orchestra can be understood if we look at some practices that are known with the two in the past. While a type of orchestra was named philharmonic or symphony orchestra in the past, the second type was named chamber orchestra.
Philharmonic or symphony orchestras usually have a larger number of players compared to chamber orchestra that is limited to about 50 players. However, the little difference between philharmonic and symphony orchestra is that people used to pay to get their tickets before they could watch the symphony orchestra, while philharmonic orchestra could be viewed for free.