Aside from the obvious content increase (the PHB has more spells, classes, and races than the Basic Rules do), what other rules changes/additions/ options exist in the PHB that are not in the Basic Rules?
NOTE: This answer assumes that the asker is comparing the latest version of the Player's Basic Rules (v0.2 at time of writing) to the Player's Handbook.
Here is a complete summary in list form of the new content:
- 5 New Races, 1 New Elf Sub-Race. Side Note: Unfortunately, Duergar Dwarves didn't make it to the PHB.
- 8 New Classes, Expanded Customization Options for Existing Classes.
- 7 New Backgrounds.
- Rules for Multiclassing.
- 42 Feats.
- Many New Spells.
- Additional Appendices. Side Note: PHB contains information about the various planes of existence, basic animal/monster stat-blocks, and "inspirational reading" which is basically a list of other sources for literary flavor/fluff text.
Interestingly, the latest version of the Player's Basic Rules contains some background information regarding factions that the PHB does not have (which presumably will be in the Dungeon Master's Guide). It's also available in the Adventurer's League Player's Guide.
The Dungeon Master's Basic Rules (v0.1 at time of writing) seems to have all of the animals/monsters that are in the PHB, plus additional monster content from the Starter Set adventure (Lost Mine of Phandelver). It does not appear to have the monsters from the latest official adventure (Hoard of the Dragon Queen), however, those are also freely available in a separate online supplement.