Demás y De más
Demás is a indefinite pronoun that equals to "otro, otros, otras". In certain cases, it is preceded by the articles lo, los, las.Vinieron al entierro Paquita, Paco y demás familia.
Paquita, Paco and the rest of the family came to the funeral.
Tú te irás de vacaciones la semana que viene y los demás cuando tú vuelvas.
You will go on vacation next week and the others when you come back.
De más is the combination of the preposition "de" adverb of quantity "más". It expresses the highest degree of something or an excess of something.
Es la persona de más buenos modales que conozco.
He is the most polite person I know.
No me pongas tanto azúcar en el café. Has puesto de más.
Don't add too much sugar to my coffee. You have add too much.
If the expression "de más" is preceded by the article "lo" (lo de más), it becomes the opposite expression of "lo de menos".
Lo de menos es que no digas la verdad, lo de más es que te inventes cosas.
It's less important that you don't say the truth, but more important that you don't make up things.
Aparte y A Parte
Aparte can be an adverb, an adjective, a noun or a verb.- As an adverb, indicates location
Pon esto aparte.
Put this aside.
- As an adjective, has the same meaning as "distinto" (different)
Eso es un caso aparte (diferente).
That's a different issue.
- As a noun, means "a break"
Hizo un aparte en su estudio para comer algo.
He took a break from studying to get something to eat.
- As a verb, is the 3rd person singular of the present subjunctive or imperative of the verb "apartar" (to move away, to set aside).
¡Aparte ese coche de nuestro camino, por favor!
¡Move that car out of the way!
If as an adverb it is followed by the preposition "de", together make the adverbial phrase "aparte de" that means besides.
Aparte de ser el más listo, es también el más guapo de la clase.
Besides being the most intelligent, he also is the most handsome one in class.
A parte is the combination of the preposition "a" followed by the noun "parte". It is also common to find the indefinite pronoun "ninguna" placed in between or "alguna" placed at the end.
Nos cruzamos toda la ciudad de parte a parte buscando un regalo para mi madre.
We crossed the whole city from side to side looking for a present for my mom.
Vestido así no te dejarán entrar a ninguna parte (a parte alguna).
You would not get in anywhere dressed like that.