i will try to be as concise as possible with the answer.
When you perform an ODS on spectra or orders, you generally just focus on the response at a specific operating point and extract (and animate) the deformation of the system. You do not make any assumption on what is causing that deformation, you simply want to visualize it.
On the other hand, when you perform OMA, you try to identify a modal model, as you do in experimental modal analysis. The difference is that, since you are not measuring the input, you have to make assumptions on it: if these assumption are valid, then you will be able to correctly identify modes out of your data.
Obviously, if you excite a system at resonance it will move according to the mode shape at that resonance, so the ODS and mode shapes will look alike. However, if you have closely-space modes, very high modal density, or dyour system is damped, it might be difficult to understand what is a mode and what not just from the spectra. Modal analysis helps in making the process more reliable, and if you measure properly will also provide an estiamation of the operational damping.
Finally, another advantage of OMA is that modal parameters can be more confidently used to validate numerical models. In many cases, operational models are also avaialble, or the structure is so big that it would be extremely difficult to excite it with a known force. ODS can also be used, but as they depend on the excitation and operating conditions (a modal model does not) they provide a less objective correlation criterion.
Hope this help in asnwering your question anc clarify the difference