Evelyn, we call them "soft sign" and "hard sign. You are correct in this. We still consider them letters:) With names like: "soft sign". No problem: for example, й represents sound /j/ but is called 'ee short' (as a child I preferred to call it 'yee' while my friend's husband teaches their daughter name 'eey').
Before the revolution most letters had meaningful words starting with this letter as their name... save a few, among them х херъ, ъ еръ, ь ерь and ы еры. (I have no idea where this -ер- came from. "Хер" is still used as an euphemistic for the most common Russian swearing word starting with x. Also there is a verb похерить - to cancel (imagine crossing a document with X shape) - some modern Russians think it is a swearing word:))
In a distant past Ь and Ъ denoted sounds, short i and u. They still could sound like this 1000 years ago, judging by loans in Finnish. Later they merged with е and o - or disappeared in other positions. Бьрьвьнo->Бревно.
Before the revolution Russian wrods couldn't end with a consonat: whatever ended with a hard consondant had -ъ in the end, reminding of the times when Slavic lnagauges sounded a bit like Japanese: all the syllables open. After the revolution they tried to abandon ъ, substituting it with an apostrophe. Such spelling still occurs, for example when we type in Latin transliteration, but as you see ъ is still here. "Psychologically" they are letters for us, though not sounds.
No less than --o- and -e in 'people', or -h- in ch-:)
Functionally... in a sense their sounds are still here. "Softening' is in a sense mixign of consonant's articulation with и articulation. Especially with labials, like m. Lips are making m-sound, while tongue is in the position for some vowel close to Russian и/е, ready to make a quick transition to the vowel and influencing the timbre.