I play bass myself, started off with guitar of course, and I must say that both instruments are very different when it comes to muting. While a guitar, depending on the genre or the part you're playing (like harmonic parts and ringing notes), can sound quite good without muting... A bass often does not.
The reason for that is that the bass forms the backbone of a song, together with the drums. A song with a muddy or imprecise foundation sounds not as good as a tight bass-tone locked in with the drums (so played when it needs to be played and muted when it does not). If you compare the playing style of a guitarist and a bassist, the bassist often uses one tone at the time. The guitar player uses more harmonic chords or powerchords, so hearing more than just what's being played doesn't matter that much.
It also is very important to mute your bass correctly because of the string gauge. Bass strings are much thicker than guitar strings. So physically, the vibrations when playing a string are much stronger, so the strings next to the one being played are moving too. Therefore it is, especially for a down-tuned bass, crucial to mute your strings correctly. Even if you just play one string, the others will be moved by the sound vibes as well and are usually out of tune, compared to the tone you're playing.
However, don't be too obsessed with muting. As long as you keep your playing style clean and mute consciously whenever there's a risk of you smashing more than one string, you shouldn't have any problems with the tone. Keep an eye on it, practice a bit and you'll be fine.