ng-bootstrap and ngx-bootstrap are two different projects by two different project teams that are trying to accomplish more or less the same thing - allowing you to use Bootstrap in Angular (2+) without the use of jQuery.
They are both rebuilding the Bootstrap components using only Angular (no jQuery).
The main differences are around which version of Bootstrap they support.
- ngx-bootstrap supports Bootstrap 3 and 4.
- ng-bootstrap supports Bootstrap 4 and requires Angular 5+.
This means that if you need to use Bootstrap version 3, then ngx-bootstrap is your only real option of the two. If you can use Bootstrap 4, then you can pick between the two projects.
The other (potentially significant) difference is the teams behind the projects.
The key point to note in this regard is that the team behind ng-bootstrap was also responsible for angular-ui-bootstrap - the AngularJS (i.e. 1.x) version of the Bootstrap library.