Not a lot. The only real difference would be that the meaning of 'she began playing' would depend on the context. It could be "she began playing a song" (10 minutes ago) or "she began playing the xylophone" (10 years ago) - "she began to play" would normally mean the first option.

Thank you!

These are essentially the same. "Playing" and "to play" are both used here in the same way, as the object of the verb. "Playing" here is a gerund, and "to play" is an infinitive, and both are used here as nouns.