Ranked in these QuestionsQuestion Ranking
Pro Modern UI
Pro Free software/open source
The source code is available under the GPLv3.
Pro Very customizable
The user can add and arrange his/her own tabs with specific views and filters.
Pro Supports custom API
It supports API customization, so you don't need to worry about API limits at all.
Pro Works as expected
Pro Handles Mastodon, Fanfou and StatusNet too
You need only one app to read Twitter and Mastodon.