Here's a comprehensive list of all changes. Replace "drupal" with "backdrop" (or case-equivalent) where expected:

Public functions:

The ~250 functions from https://gist.github.com/quicksketch/7077150 all wrapped individually, including class methods.

Private functions:

No replacements provided for private functions.

_drupal_decode_exception
_drupal_exception_handler
_drupal_log_error
_drupal_error_handler
_drupal_error_handler_real
_drupal_schema_initialize
_drupal_shutdown_function
_drupal_default_html_head
_drupal_http_use_proxy
_drupal_css_aggregate
_drupal_build_css_path
_drupal_flush_css_js
_drupal_html_to_mail_urls
_drupal_session_open
_drupal_session_close
_drupal_session_read
_drupal_session_write
_drupal_session_destroy
_drupal_session_garbage_collection
_drupal_session_delete_cookie

Variables:

The "drupal_" prefix on all of these are just dropped, since they universally are not needed and are non-public anyway. They'll also be converted to config before release.

drupal_css_cache_files => css_cache_files
drupal_stale_file_threshold  => stale_file_threshold
drupal_js_version_query_string => js_version_query_string
drupal_js_cache_files => js_cache_files
drupal_private_key => private_key
drupal_weight_select_max => weight_select_max

Constants:

Below aliased directly to BACKDROP_ equivelents.*

DRUPAL_ROOT
DRUPAL_BOOTSTRAP_CONFIGURATION
DRUPAL_BOOTSTRAP_PAGE_CACHE
DRUPAL_BOOTSTRAP_DATABASE
DRUPAL_BOOTSTRAP_VARIABLES
DRUPAL_BOOTSTRAP_SESSION
DRUPAL_BOOTSTRAP_PAGE_HEADER
DRUPAL_BOOTSTRAP_LANGUAGE
DRUPAL_BOOTSTRAP_FULL
DRUPAL_ANONYMOUS_RID
DRUPAL_AUTHENTICATED_RID
DRUPAL_KILOBYTE
DRUPAL_PHP_FUNCTION_PATTERN
DRUPAL_MINIMUM_PHP
DRUPAL_MINIMUM_PHP_MEMORY_LIMIT

Below manually redefined because their files are not loaded early enough.

// common.inc
const DRUPAL_NO_CACHE = -1;
const DRUPAL_CACHE_CUSTOM = -2;
const DRUPAL_CACHE_PER_ROLE = 0x0001;
const DRUPAL_CACHE_PER_USER = 0x0002;
const DRUPAL_CACHE_PER_PAGE = 0x0004;
const DRUPAL_CACHE_GLOBAL = 0x0008;

// password.inc
const DRUPAL_HASH_COUNT = 16;
const DRUPAL_MIN_HASH_COUNT = 7;
const DRUPAL_MAX_HASH_COUNT = 30;
const DRUPAL_HASH_LENGTH = 55;

// system.module
const DRUPAL_MAXIMUM_TEMP_FILE_AGE = 21600;
const DRUPAL_CRON_DEFAULT_THRESHOLD = 10800;
const DRUPAL_USER_TIMEZONE_DEFAULT = 0;
const DRUPAL_USER_TIMEZONE_EMPTY = 1;
const DRUPAL_USER_TIMEZONE_SELECT = 2;
const DRUPAL_DISABLED = 0;
const DRUPAL_OPTIONAL = 1;
const DRUPAL_REQUIRED = 2;
const DRUPAL_WEIGHT_SELECT_MAX = 100;

Globals:

For testing only and not public, so it's not wrapped.

drupal_test_info => backdrop_test_info

Classes and Interfaces:

All wrapped with compatibility mode.

DrupalQueue
DrupalQueueInterface
DrupalReliableQueueInterface
DrupalCacheArray
DrupalUpdateException
DrupalFakeCache
DrupalDatabaseCache
DrupalNullCache
DrupalLocalStreamWrapper
DrupalPublicStreamWrapper
DrupalPrivateStreamWrapper
DrupalTemporaryStreamWrapper
DrupalStreamWrapperInterface
DrupalUpdaterInterface
DrupalDefaultEntityController
DrupalEntityControllerInterface

Javascript:

All wrapped in compatibility mode.

Drupal.behaviors.*
Drupal.attachBehaviors()
Drupal.settings.*
Drupal.ajax.*
Drupal.t()
etc.

Libraries:

Using drupal_add_library() with drupal.* works with compatibility mode.

drupal.ajax => backdrop.ajax
drupal.batch => backdrop.batch
etc.
Introduced in branch: 
1.0.x
Introduced in version: 
1.0.0
Impacts: 
Module developers
Theme developers
Related Github Issues: 
https://github.com/backdrop/backdrop-issues/issues/3