A common problem.
CMYK print processes cannot produce as bright and colorful looking colors as the usual sRGB screen. Some programs can make usefully accurate predictions on the screen what CMYK printing would produce. Adobe products and CorelDraw have got their position in the industry because that capability. There are some free and low cost alternatives which also understand CMYK printing: Krita, Scribus, Serif Affinity series. But GIMP knows nothing of it.
To get reliable results for ex. in Photoshop you must use right RGB and CMYK color profiles which know how your monitor and how the forthcoming CMYK printing handle color. Unfortunately low cost office printers do not have such profiles, they present themselves to Photoshop as perfect sRGB capable printers which is of course a lie and the result will be something like in your 2nd image.
Learn the color management concept from the numerous available sources, for ex. Adobe's manuals. Design the colors of your art so that the colors are printable. If that's done properly the result can still look good as you can see in magazines.
In your current case the image obviously has several layers. Transparent layers in CMYK are combined differently with the background in CMYK images than in RGB images. The suggestion given in a comment "flatten the image before converting it to CMYK" is good.
See the next image which has a little too saturated red on brown. The red cannot be shown properly in the used CMYK printing color profile which makes it as dull as in your image.
In the left there are the same shapes as in the right, but flattened before converting the image from RGB to CMYK. In the right the glow is a separate layer.
The glow suffers baldly due the different layer mixing priciple. In RGB the transparent glow adds color on the background, in CMYK it takes color off. The result becomes even worse if you use some layer blending mode other than Normal, for ex. Screen which is common in glow effects.
ADD in May 2022: GIMP 2.10 understands CMYK print profiles, it can show the CMYK printing result and can show a warning if something is unprintable (=out of gamut) in CMYK. This all is availabe in the real time when one edits his RGB image in GIMP. The actual conversion to CMYK must still be done outside GIMP, for ex. in Krita, if one wants to use Freeware.
I need your help. I am trying to convert my RGB poster into a printable CMYK poster. But when I change the "neon like" headline to CMYK colours and the document color mode in illustrator the Blur effects becomes grey. The same thing goes with the blur and transparent effect in indesign when I transform the document and colours to CMYK. The background picture is converted to CMYK also, but when I activate proof setup at proof colors, the colors becomes flat like a grey overlay was added.