Well, they're both correct sentences. And the difference between them is really small. But here it comes:
- ESTABA is an IMPERFECTIVE tense, these tenses are used to speak about actions in progress, processes that have a duration in time.
- ESTUVO es a PERFECTIVE tense, these tenses are used to speak about actions that are completed, or have no duration in time (just an instant)
For example: Ayer comí (Yesterday I ate), vs. Ayer comía (yesterday I was eating). Normally you just want to say that you ate, or what you ate, so you´d use the perfective tense, but if you want to speak about something that happened while you were eating, then you´d say "ayer mientras comía, estaba oyendo la radio).
In your example, I think that 90% of the time people would say "ella estuvo aprendiendo" because it´s something that is already done, finished, completed successfully. But, if you wanted to talk about something that happened while she was there, you might say "la conocí cuando ella estaba aprendiendo a tocar el contrabajo"