Assent and ascent are two soundalike words that are just uncommon enough that we might stumble over the spellings. The sc combination in the one that involves climbing mimics the first two letters of scale, which as a verb means to climb something. The words ascent and scale are ultimately related through the Latin scandere, to climb.
To give your assent means you are agreeable. Assent is an indication that you are going along with the plan, and not being a stubborn ass.
The c in ascent is either pronounced like an s or it’s silent, as in scent. The similar accent has a different pronunciation that use the k sound and then the s sound. Thankfully for learners of English, acsent is not a word, although it does show up as a typo for either accent or ascent at times.
Assent and scent are related words, and both are related to sense and sentient. The root is the Latin sentire, or feel, as in perceive. Ascent shows up in the last 16th century based on the idea that if you have a noun form, descent, with the verb, descend, you should also have an ascent to go with ascend. If ascent were to appear today, it would be derided as a “made up” word.
That c in scent is probably just a very old misspelling. It doesn’t really belong there. The noun was preceded by a 15th century verb that meant to perceive, and then the meaning was narrowed to describe how hunting dogs perceive prey. It's uncommon, but we still sometimes scent something out, literally or figuratively. The scent spelling doesn’t show up regularly until the late 17th century.