Apps is way more convenient, faster to access, and keeps users up to date with notifications. If you think that your user will be playing with your app a few times a day, then it makes more sense to go with it first. But still website is an important thing to develop it then.
One of the reasons you should have a website is the full-time availablity 24/7 just think of employing a hall customer services for 24/7 all the year with no weekends or breaks! this is one of the costs you can save today by making your website, Website is important for you / your business even if your business is not focusing on technology.
Facing any problem can make you cost lots of money and take time finding freelancer/worker/company to solve your problem, as they will take more time checking how your system was developed to start solve the problem with out causing other problems. Technical Support are available for you to be ready solving any problem you may face as they will also make some system improvments monthly wich is a benfit for less yearly costs.
API is the shortcut of "Application Programming Interface" which is the gateway between the programmers and the system, It's very important in Mobile Apps as of multiple advantages one of them is protection where using the API for both application & website makes it more powerful and safer as if there were any problem in the Base System, changing it from the API makes it protected for all platforms uses your system.
If you've heard "UX" anywere before, be aware that "UX" is a general word but it used most in technology such as Apps, Websites and Video Games. "UX" is the shortcut of "User Experience" which describes how and what is the best way (for the target group who will use the app/game/website) to interact with it in details. ex: Play button should be in the middle of the screen because the game is a Clicker-Game Type.
Frontend is the design, interactions, effects which be processed in the client side (visitor of website/app) which is the viewed side of the project whether it is an app or website. Backend is the system which working in the background side includes contacting to the data on database, calculating and algorithms solving, check the user details, send emails and much more... .