Playing podcasts via either service works pretty much the same way. You'll be able to use voice commands like "Alexa, play The Joe Budden Podcast on Spotify," or "Alexa, play The Daily from yesterday on Apple Podcasts." You can also tell Alexa to do things like fast forward or play the next episode via both apps.
Plus, if you pause a podcast in either the Apple Podcasts or Spotify apps on your phone, you can then tell Alexa to resume playing it on your smart speaker. If you want, you can set either Apple Podcasts or Spotify as your preferred podcast provider in the Alexa app. Go to Settings -> Music & Podcasts -> Default Services, then select one of the apps you have lined to Alexa.