"Ь" is a soft sign, it appears only after cosonants and "softens" them. It can't be ignored, because it affects both pronunciation and meaning. For example, мел = chalk, мель = sandbank (and they are pronounced differently). There are no words starting with "ь" (as there nothing to soften for it if it's the first letter).

"Ы" is NOT a soft sign, it's a vowel. That means that it's an actual sound that can be heard.

You probably confused it with "Ъ" which is a "hard sign". The function of "ъ" is very limited in modern Russian, it's remnants of old spelling. Similar to the soft sign it appears only after consonants and affects pronunciation of the following vowel. Most often you will see it between a prefix and a root in a number of words. For example, "съезд" (due to ъ it's pronounced as [сйезд])