You can see the error, if there’s one, by looking in the browser’s console. This is how to have a look in your browser’s console window: https://webmasters.stackexchange.com/a/77337
Also, another option is to contact the theme developer and to request some help. This is a short article explaining how you can contact the developer: https://help.market.envato.com/hc/en-us/articles/203039054-How-to-contact-an-author
The difference between having a map displayed using an iframe or using the Maps API is that using an iframe, you can show some locations on the map, but you can’t interact too much with the map. While using the Maps API, you can have a more dynamic map, you can interact more with it (search for locations, calculate distances, enable/disable controls on the map, apply styles and other things)
A very important thing is that if you want to use the Maps API, Google requires an API key. Most probably, the documentation of the theme you purchased explains how to get the API key.
All the best!