と共に|とともに

1. 1Together with; along with (a person, group, or thing).Ex. 1
2. 2As X changes, Y also changes along with it.Ex. 2
3. 3At the same time as an event, or as well as being something (both things hold).

A somewhat formal expression common in writing, news, speeches, and formal explanations. In everyday conversation, と一緒に is more natural for 'together with.' Unlike につれて, which expresses only proportional change, と共に also covers accompaniment and simultaneity.

• に伴ってN2「に伴って」 is close to 「と共に」 in two senses — 'as something changes' and 'a second situation arising alongside some event' — and the two are often interchangeable. It cannot, however, express 'together with' or 'having two qualities at once'; in official announcements, 「に伴い」 is commonly used.
• につれてN3「につれて」 corresponds only to 「と共に」's 'changing along with something' sense, and it stresses gradual, continuous change; its register is neutral, and it is common in speech as well. It cannot be swapped in for 'together with,' 'happening at the same time,' or 'having two qualities at once.'
• にしたがってN3「にしたがって」 is interchangeable with 「と共に」 only in the 'as something changes' sense; its register is neutral, and it is less formal than 「と共に」. It also has a separate 'in accordance with / following' sense, but it does not express 'together with,' 'at the same time,' or 'having two qualities at once.'
• と同時にN2「と同時に」 overlaps with 「と共に」 only in the 'happening at the same time' and 'having both sides' senses, with stronger emphasis on the exact same instant; its register is neutral, and it can also attach to a verb's た-form. It cannot express 'together with' or 'changing along with something.'
• ながら / いながら / ながらもN3「ながら」 attaches to the ます-stem and expresses the same subject doing A while doing B (the focus falls on the second action), and it can also be concessive, 'although...'; its register is neutral. 「と共に」 leans written, has no concessive use, and is not used for the everyday 'doing A while doing B.'

「と共に」 has the fullest range of senses — 'together with,' 'as something changes,' 'at the same time,' and 'having two qualities at once' — and leans toward written style. The others each cover only part of this: 「に伴って」「につれて」「にしたがって」 overlap mainly in the 'as something changes' sense, with 「につれて」 stressing gradual, continuous change and 「に伴って」 also covering a second situation arising alongside some event; 「と同時に」 matches only the 'same time / two qualities' senses, with stronger emphasis on the exact same instant; none can replace 「と共に」 for 'together with,' and 「ながら」's 'doing A while doing B' and concessive senses don't use 「と共に」 either.