Their feature sets are not at all the same. Sqlite is an embedded database which has no network capabilities (unless you add them). So you can't use it on a network.
If you need
- Network access - for example accessing from another machine;
- Any real degree of concurrency - for example, if you think you are likely to want to run several queries at once, or run a workload that has lots of selects and a few updates, and want them to go smoothly etc.
- a lot of memory usage, for example, to buffer parts of your 1Tb database in your 32G of memory.
You need to use mysql or some other server-based RDBMS.
Note that MySQL is not the only choice and there are plenty of others which might be better for new applications (for example pgSQL).
Sqlite is a very, very nice piece of software, but it has never made claims to do any of these things that RDBMS servers do. It's a small library which runs SQL on local files (using locking to ensure that multiple processes don't screw the file up). It's really well tested and I like it a lot.
Also, if you aren't able to choose this correctly by yourself, you probably need to hire someone on your team who can.