What are the practical differences between Bash and Zsh?
First, some important things:


Bash isn't going away. If you're already using bash, nothing will change for you. All that changes is that zsh will be the default login shell for new accounts, and even then, you can select bash instead.
Scripts are not affected. What changes is the shell for interactive use, i.e. the shell in terminals (and also a few other things that use the login shell, such as crontabs). If you have a script in a file with execution permissions, starting with a shebang line such as #!/bin/bash or #!/bin/sh or #!/usr/bin/env bash, it'll keep working exactly as before.
Zsh's syntax is not completely compatible with bash, but it's close. A lot of code will keep working, for example typical aliases and functions. The main differences are in interactive configuration features.


Now, assuming you're considering switching to zsh, which has been a possibility for years, here are the main differences you'll encounter. This is not an exhaustive list!

Main differences for interactive use

Configuration files: bash reads (mainly) .bashrc in non-login interactive shells (but macOS starts a login shell in terminals by default), .profile or .bash_profile in login shells, and .inputrc. Zsh reads (mainly) .zshrc (in all interactive shells) and .zprofile (in login shells). This means that none of your bash customizations will apply: you'll need to port them over. You can't just copy the files because many things will need tweaking.

Key bindings use completely different syntax. Bash uses .inputrc and the bind builtin to bind keys to readline commands. Zsh uses the bindkey builtin to bind keys to zle widgets. Most readline commands have a zsh equivalent, but it isn't always a perfect equivalence.

Speaking of key bindings, if you use Vi(m) as your in-terminal editor but not as your command line mode in the shell, you'll notice zsh defaults to vi editing mode if EDITOR or VISUAL is set to vi or vim. bindkey -e switches to emacs mode. 

Prompt: bash sets the prompt (mainly) from PS1 which contains backslash escapes. Zsh sets the prompt mainly from PS1 which contains percent escapes. The functionality of bash's PROMPT_COMMAND is available in zsh via the precmd and preexec hook functions. Zsh has more convenience mechanisms to build fancy prompts including a prompt theme mechanism.

The basic command line history mechanisms (navigation with Up/Down, search with Ctrl+R, history expansion with !! and friends, last argument recall with Alt+. or $_) work in the same way, but there are a lot of differences in the details, too many to list here. You can copy your .bash_history to .zsh_history if you haven't changed a shell option that changes the file format.

Completion: both shells default to a basic completion mode that mostly completes command and file names, and switch to a fancy mode by including bash_completion on bash or by running compinit in zsh. You'll find some commands that bash handles better and some that zsh handles better. Zsh is usually more precise, but sometimes gives up where bash does something that isn't correct but is sensible. To specify possible completions for a command, zsh has three mechanisms:


The “old” completion mechanism with compctl which you can forget about.
The “new” completion mechanism with compadd and lots of functions that begin with underscore and a powerful but complex user configuration mechanism.
An emulation to support bash completion functions which you can enable by running bashcompinit. The emulation isn't 100% perfect but it usually works.


Many of bash's shopt settings have a corresponding setopt in zsh.

Zsh doesn't treat # as a comment start on the command line by default, only in scripts (including .zshrc and such). To enable interactive comments, run setopt interactive_comments.

Main differences for scripting

(and for power users on the command line of course)

In bash, $foo takes the value of foo, splits it at whitespace characters, and for each whitespace-separated part, if it contains wildcard characters and matches an existing file, replaces the pattern by the list of matches. To just get the value of foo, you need "$foo". The same applies to command substitution $(foo). In zsh, $foo is the value of foo and $(foo) is the output of foo minus its final newlines, with two exceptions. If a word becomes empty due to expanding empty unquoted variables, it's removed (e.g. a=; b=; printf "%s\n" one "$a$b" three $a$b five prints one, an empty line, three, five). The result of an unquoted command substitution is split at whitespace but the pieces don't undergo wildcard matching.

Bash arrays are indexed from 0 to (length-1). Zsh arrays are indexed from 1 to length. With a=(one two three), in bash, ${a[1]} is two, but in zsh, it's one. In bash, if you just reference an array variable without braces, you get the first element, e.g. $a is one and $a[1] is one[1]. In zsh, $a expands to the list of non-empty elements, and $a[1] expands to the first element. Similarly, in bash, the length of an array is ${#a}; this also works in zsh but you can write it more simply as $#a. You can make 0-indexing the default with setopt ksh_arrays; this also turns on the requirement to use braces to refer to an array element.

Bash has extra wildcard patterns such as @(foo|bar) to match foo or bar, which are only enabled with shopt -s extglob. In zsh, you can enable these patterns with setopt ksh_glob, but there's also a simpler-to-type native syntax such as (foo|bar), some of which requires setopt extended_glob (do put that in your .zshrc, and it's on by default in completion functions). **/ for recursive directory traversal is always enabled in zsh.

In bash, by default, if a wildcard pattern doesn't match any file, it's left unchanged. In zsh, by default, you'll get an error, which is usually the safest setting. If you want to pass a wildcard parameter to a command, use quotes. You can switch to the bash behavior with setopt no_nomatch. You can make non-matching wildcard patterns expand to an empty list instead with setopt null_glob.

In bash, the right-hand side of a pipeline runs in a subshell. In zsh, it runs in the parent shell, so you can write things like somecommand | read output.

Some nice zsh features

Here are a few nice zsh features that bash doesn't have (at least not without some serious elbow grease). Once again, this is just a selection of the ones I consider the most useful.

Glob qualifiers allow matching files based on metadata such as their time stamp, their size, etc. They also allow tweaking the output. The syntax is rather cryptic, but it's extremely convenient. Here are a few examples:


foo*(.): only regular files matching foo* and symbolic links to regular files, not directories and other special files.
foo*(*.): only executable regular files matching foo*.
foo*(-.): only regular files matching foo*, not symbolic links and other special files.
foo*(-@): only dangling symbolic links matching foo*.
foo*(om): the files matching foo*, sorted by last modification date, most recent first. Note that if you pass this to ls, it'll do its own sorting. This is especially useful in…
foo*(om[1,10]): the 10 most recent files matching foo*, most recent first.
foo*(Lm+1): files matching foo* whose size is at least 1MB. 
foo*(N): same as foo*, but if this doesn't match any file, produce an empty list regardless of the setting of the null_glob option (see above).
*(D): match all files including dot files (except . and ..).
foo/bar/*(:t) (using a history modifier): the files in foo/bar, but with only the base name of the file. E.g. if there is a foo/bar/qux.txt, it's expanded as qux.txt.
foo/bar/*(.:r): take regular files under foo/bar and remove the extension. E.g. foo/bar/qux.txt is expanded as foo/bar/qux.
foo*.odt(e\''REPLY=$REPLY:r.pdf'\'): take the list of files matching foo*.odt, and replace .odt by .pdf (regardless of whether the PDF file exists).


Here are a few useful zsh-specific wildcard patterns.


foo*.txt~foobar*: all .txt files whose name starts with foo but not foobar.
image&lt;-&gt;.jpg(n): all .jpg files whose base name is image followed by a number, e.g. image3.jpg and image22.jpg but not image-backup.jpg. The glob qualifier (n) causes the files to be listed in numerical order, i.e. image9.jpg comes before image10.jpg (you can make this the default even without -n with setopt numeric_glob_sort).


To mass-rename files, zsh provides a very convenient tool: the zmv function. Suggested for your .zshrc:

autoload zmv
alias zcp='zmv -C' zln='zmv -L'


Example:

zmv '(*).jpeg' '$1.jpg'
zmv '(*)-backup.(*)' 'backups/$1.$2'


Bash has a few ways to apply transformations when taking the value of a variable. Zsh has some of the same and many more. 

Zsh has a number of little convenient features to change directories. Turn on setopt auto_cd to change to a directory when you type its name without having to type cd (bash also has this nowadays). You can use the two-argument form to cd to change to a directory whose name is close to the current directory. For example, if you're in /some/where/foo-old/deeply/nested/inside and you want to go to /some/where/foo-new/deeply/nested/inside, just type cd old new.

To assign a value to a variable, you of course write VARIABLE=VALUE. To edit the value of a variable interactively, just run vared VARIABLE.

Final advice

Zsh comes with a configuration interface that supports a few of the most common settings, including canned recipes for things like case-insensitive completion. To (re)run this interface (the first line is not needed if you're using a configuration file that was edited by zsh-newuser-install):

autoload -U zsh-newuser-install
zsh-newuser-install


Out of the box, with no configuration file at all, many of zsh's useful features are disabled for backward compatibility with 1990's versions. zsh-newuser-install suggests some recommended features to turn on.

There are many zsh configuration frameworks on the web (many of them are on Github). They can be a convenient way to get started with some powerful features. The flip side of the coin is they often lock you in doing things the way the author does, so sometimes they'll prevent you from doing things the way you want. Use them at your own risk.

The zsh manual has a lot of information, but it's often written in a way that's terse and hard to follow, and has few examples. Don't hesitate to search for explanations and examples online: if you only use the part of zsh that's easy to understand in the manual, you'll miss out. Two good resources are the zsh-users mailing list and Unix Stack Exchange.
An extensive collection of articles on switching to zsh on the mac can be found on scriptingosx.com and a useful Ruby script to bring your command history with you, can be found on Github.