Ring dips and ring push-ups definitely overlap; however, it will be beneficial to continue the push-ups after you able to do dips for a couple of reasons.
- Dips work your triceps more, while push-ups work the chest more.
- You will continue to be able to do more push-ups even when you can do dips, so you can start with dips and finish your work-out with push-ups.
- Variety produces adaptation. Doing both will give you more gains than only doing one.
- According to Ring Training for Elite Fitness (Tyler Hass), though the two exercises are both "pushes", it is useful to alternate between them. This is because they operate in different planes: vertical (ring dips) and horizontal (ring push-ups).
A progression between the push-up and the dip would be to be vertical and use your legs a little bit for assistance (or get a rubber band to support some of your weight). Also, a similar exercise, which uses your shoulders more and is a progression for ring handstand push-ups, would be raising your legs a little bit above the rings.