1. Convert and split animated GIF into PNG sequence
convert -coalesce something.gif something.png
or
magick convert -coalesce something.gif something.png
convert/magick convert is a command line tool from the ImageMagick.org software package. The command name depends on the software version.
This produces a set of numbered PNG files something-0.png, ..., something-16.png (Here, the original GIF http://i.stack.imgur.com/VHJmL.gif, renamed to something.gif consists of 17 frames.) Option -coalesce is necessary to undo a possible optimization of the original GIF file.
2. Get original animation speed
magick identify -verbose something.gif | grep 'Delay'
Users of Windows may want to run
magick identify -verbose something.gif | sls -Pattern 'Delay'
in the PowerShell.
This outputs lines (one for every frame) like:
Delay: 10x100
Delay: 10x100
Delay: 10x100
...
The frame rate (frames per second), to be passed as an argument to the \animategraphics command below, is found by dividing the number after x by the number in front of it:
frame rate = 100 (tics/s) / 10 (tics/frame) = 10 frames/s
3. Embed PNG sequence as an embedded looping animation in the final PDF
(This kind of animation requires a JavaScript-supporting PDF viewer, such as Acrobat Reader or KDE Okular.)
\documentclass{beamer}
\usepackage{animate}
\begin{document}
\begin{frame}{Embedded Animation}
\animategraphics[loop,controls,width=\linewidth]{10}{something-}{0}{16}
\end{frame}
\end{document}
Argument {10} sets the desired frame rate (frames per second), {0} and {16} set the first and last file numbers of the PNG series to be included in the animation. Note that frame rates above 30 FPS, if at all achieved by the PDF viewer, do not make much sense. 30 FPS is a typical value in video encoding. Use command option measure and the + button to see which frame rates are possible. They may depend on the image size and of course on the hardware on which the PDF viewer is running.
For a more GIF-like impression, option autoplay can be used instead of or in addition to option controls.
Best Answer
PDF is not just designed for printing. It is a format for displaying electronic documents, independently of hardware (printer, screen, e-reader, smartphone, projector etc.) and software (such as the operating system). This format is both good for printouts and presentation, so a good choice as output format.
Presentations with technical or scientific content are often based on documents, very often written in LaTeX also because of it's mathematical typesetting capabilities. It's very natural to use the same tool, LaTeX, for the presentation.
The benefits of LaTeX, such as separating the form/style and the content, portability in source, implementation and output, cross-referencing capabilities and typesetting quality, are great for presentations as well.
As I use LaTeX, I can work on the source using Windows, Linux, or Mac OS X, and I can give the presentation away as a PDF for reading or presenting on a conference, the reader or speaker can use Windows, Linux, Mac OS X, a tablet computer, a smartphone - or a printout.
If I would use PowerPoint - well, I would be pretty bound to Windows Version X with PowerPoint Version Y or a suited viewer, which is also a limit for the reader or presenter.