Using Google Places API and storing the results in your own database likely violates the terms of service. Likewise, you are limited to the number of calls per day to the API. If you expect users to appreciate (and pay for) the software you are generating, you should do likewise. Go with the licenced database.
According to the Google Places API license agreement you can only "store" or cache data for a maximum of 30 days and there are additional restrictions around the use of that data (see excerpt below). I think a better explanation of how long the data is going to need to be stored and how it is going to be used needs to be offered in order to determine if Google Places API is even a viable option.
Disclaimer: Please note any discussion around licensing requires both legal and technical expertise. I am not an attorney and can only offer my technical advice. The advice given in this answer is NOT legal advice. You should also consult your attorney about the legal ramifications surrounding this question.
Excerpt taken from ( https://developers.google.com/maps/terms ) on 20-May-2015
"10.1.3 Restrictions against Copying or Data Export.
(a) No Unauthorized Copying, Modification, Creation of Derivative Works, or Display of the Content. You must not copy, translate, modify, or create a derivative work (including creating or contributing to a database) of, or publicly display any Content or any part thereof except as explicitly permitted under these Terms. For example, the following are prohibited: (i) creating server-side modification of map tiles; (ii) stitching multiple static map images together to display a map that is larger than permitted in the Maps APIs Documentation; (iii) creating mailing lists or telemarketing lists based on the Content; or (iv) exporting, writing, or saving the Content to a third party's location-based platform or service.
(b) No Pre-Fetching, Caching, or Storage of Content. You must not pre-fetch, cache, or store any Content, except that you may store: (i) limited amounts of Content for the purpose of improving the performance of your Maps API Implementation if you do so temporarily (and in no event for more than 30 calendar days), securely, and in a manner that does not permit use of the Content outside of the Service; and (ii) any content identifier or key that the Maps APIs Documentation specifically permits you to store. For example, you must not use the Content to create an independent database of "places" or other local listings information."