Beyond maintainence (see Angular's changelog which has hundreds of bug fixes since 1.4.5), there's a few reasons I'd like to upgrade (in order of concreteness):

1. Custom comparators for ordering elements (e.g. multi-column sorting, see jdiefenbaugh/beat.web#22)
2. Performance improvements which would help smooth the UX
3. Lets us upgrade jQuery to v3
4. (ties in with 3) lets us track the current development/maintenance of Angular 1 & its ecosystem

As far as breaking changes go - there's not much, since we're just using standard Angular features:

1. There might be an issue with $sce, Angular's utility for sanitizing strings before using them in things like resource URLs or dynamically-compiled HTML. Angular may require using $sce more often in our code, if we use complex $compile calls or use dynamic resource hrefs in our templates.
2. Angular's custom success and error methods for calls to $http have been deprecated and then removed. They now recommend using the official JS solution, then and catch, which have huge browser support & more features. There aren't many instances we need to change, either. A quick grep shows:

beat/web/ui/static/ui/js/reportdialog.js
153:                           .success(function (reportData)
160:                           .error(function (error)
250:                               .success(function (reportData)
257:                               .error(function (error)
273:                               .success(function (reportData)
280:                               .error(function (error)
394:                           .success(function (reportData)
401:                           .error(function (error)

beat/web/plotters/static/plotters/app/controllers/plotterparameterController.js
64:           .success(function (plottersData)
72:           .error(function (error)
83:           .success(function (plotterparameterData)
124:           .error(function (error)

beat/web/plotters/static/plotters/app/directives/plotterparameterItemView.js
201:                    .success(function (returnedData)
207:                    .error(function (error)
252:                    .success(function (returnedData)
257:                    .error(function (error)
314:                    .success(function (returnedData)
319:                    .error(function (error)
381:                    .success(function (returnedData)
386:                    .error(function (error)

beat/web/reports/static/reports/app/directives/saveReportItems.js
35:                     .error(function (error){