Also, top answer from that ELI5 link I posted since you won't open it:

tl;dr: GIF is an image compression format, which isn't really suited well to compressing video data. Basically the only advantage is that browsers support the format natively without loading plugins. Actual videos compressed with video encoders are way more efficient.

In an animated GIF, each frame of the animation is encoded as a separate file, and then the files are more or less just stuck together in the final animation. You can do something similar with JPEG (Motion JPEG), although it's not supported by browsers in quite the same way as GIFs.

Also GIFs do a better job of compressing line art rather than photographs. Thus, making a video out of GIF images yields pretty poor compression anyhow.

In an encoded video file, the video compressor takes into account that successive frames are (probably) related to each other. The first frame is encoded completely, in a similar (but more advanced) method as a JPEG might be. However, the next frame is typically encoded as just the difference from the first frame to the second. This is way more efficient for most videos where motion happens slowly.

Say the video is of a ball falling.

In an animated GIF, each frame is a standalone, complete picture of the ball and any surroundings. It's compressed rather poorly, and a lot of redundant information is sent, so each individual frame is very large.

In a video encoded with a video codec (MPEG1, MPEG2, MPEG4, h.264, h,265, etc), the first frame is compressed somewhat similarly to a JPEG. Then the second frame basically says "see the region of the first frame where the ball is? Move that down a bit. Fill in the background with this information." - This frame is way smaller and easier to transmit than the first frame. It's also easier and faster to decode.

There are some drawbacks to the video compression approach, but in general it does a way better job of representing the video data with way fewer bits.

EDIT: Wow, this is getting a lot of attention :) I left out a lot of stuff to keep the answer simple, but I'm getting pinged about a lot of the same things over and over, so here's a few quick additions:

Yes, compression algorithm differences matter as well. I said something to this effect, but to extend: GIF uses a compression method that works well for color patterns, and video encoders use something that works better for photographs/live video. I didn't try to explain DCT/Quantization/entropy coding since that's a super technical topic for ELI5.
Yes, GIF can represent a simple delta between two frames. However, in the case of video data, it is rare that this is useful. Someone asked for a more detailed explanation of this, my response is here.
Yes, there may be other factors at work besides the bitrate/coding issues. I think it is safe to say that GIF is a terrible video compression format, and that's probably the root of many other issues. However, I haven't profiled browsers to see where they spend time decoding animated GIFs. I made a list of other places that could be inefficient here.
Yeah, your "five year old didn't understand this" el-oh-el :P If my 7-yr-old had the interest I'd try to explain it to him and get his feedback, but I think he'd rather play LEGO.

EDIT 2: A couple more things people are asking/mentioning:

Several people have mentioned HTML5 Video and WebM as alternatives to animated GIFs. There's a lot of back-and-forth, legal issues, competitors refusing to agree and so forth that is really hampering the HTML5 video tag. Wikipedia has a long article on the subject. We've had better video technology than GIF for years and GIF is still going strong. I think we're stuck with it, despite it being a terrible technology for the problem. (See also: practically every other technology used on the web)
A common question is "So does this mean that simpler videos with less movement will be encoded to smaller file sizes? Or that complex or really random videos make larger or less efficient video files?" The answer to this somewhat depends on the encoder settings, but in general the answer is yes. If you encode a simple video with little motion it will take less data than a more complex video with a lot of motion and random stuff going on. Modern video encoders are tuned towards encoding video sequences from cameras. They make assumptions based on what the real world looks like most of the time, and use those assumptions to compress more efficiently. They also take into account what our eyes are good at seeing.