What a GIF holds, and what the PNG gets
A GIF is a picture of at most 256 colours, compressed with LZW, with a single transparent colour and, optionally, several frames. MDN lists exactly that, and it advises PNG for lossless still pictures with an indexed palette. So the PNG is the natural still version of a GIF.
The conversion keeps the picture as it is. We compared three kinds of GIF with Pillow's first frame: one with a transparent colour, one interlaced, and animated ones with 5 and 12 frames. In every case the PNG had the same pixels, and the number of transparent pixels (4,220 in the transparent test file) was identical.
The PNG can hold more than a GIF can, such as millions of colours and soft transparency, but a converted GIF does not use that room. It still has at most 256 colours and the same hard edges. We counted: the PNG held exactly the colours of the GIF, 256 in a photo, 4 in a flat logo and 3 in a logo on a transparent background. A PNG with 256 colours or fewer is stored as a palette picture, and one made only of greys as 8 bit greyscale, which is why the files stay small.
Interlacing does not matter. An interlaced GIF stores its rows in an odd order so that a browser can show a rough picture early. The picture is the same, and our interlaced test file gave the same pixels as Pillow's reading of it.
The first frame only
An animated GIF is many pictures in one file. The converter takes the first and ignores the rest. When a file has more than one frame, the row says "animated, first frame only" next to its size, so a batch cannot silently lose animations.
That first frame is not always the best still. Some GIFs begin with a blank or partial frame, and a looping GIF has no single "main" picture. If you need another moment from a clip, the better source is the video it came from, and Video to GIF can make a new GIF from any part of a video. This page does not choose frames.
Transparent GIFs and hard edges
A GIF pixel is either fully transparent or fully opaque. Artists work around this by blending the edge of a shape into a background colour, which leaves a visible fringe when the graphic is later placed on another colour. Converting to PNG does not remove that fringe, because the fringe is made of opaque pixels in the picture itself.
If a graphic with a white halo is going onto a dark page, the fix is in an image editor, with the original artwork. Converting formats cannot undo it.
Size and when not to bother
How much smaller depends on the picture. Flat artwork shrinks most: a 400 by 300 four colour logo went from 2.1 KB to 887 bytes (58 percent smaller), the same logo on a transparent background from 1.6 KB to 606 bytes (63 percent), and a smooth 600 by 200 gradient from 21.6 KB to 483 bytes (98 percent). A photograph squeezed into 256 colours shrinks least: a 640 by 480 one went from 212.6 KB to 172 KB, only 19 percent, because the dots that fake the missing colours do not compress well. A transparent test GIF became a PNG 35 percent smaller, and the first frame of a 5 frame animation became a PNG 86 percent smaller than the whole GIF, since the PNG holds one frame and the GIF holds all of them.
If you only want a still for a website, the format that gives the smallest file for photographic frames is not PNG. Photographic first frames are usually smaller as a JPG, and GIF to JPG covers that. For graphics, logos and screenshots, PNG is the better still.