Menor vs. mais pequeno
Both menor and mais pequeno translate to "smaller" in English, but their usage depends heavily on grammatical rules and regional dialects. Generally, menor is the irregular comparative form (like saying "better" instead of "more good"), while mais pequeno is the analytic form. The choice between them often signals whether the speaker is from Brazil or Portugal.
Menor
A2This is the irregular comparative form of the adjective pequeno. In Brazilian Portuguese, menor is the standard word used for specific physical size, abstract importance, numbers, and age. In European Portuguese (Portugal), menor is typically reserved for abstract concepts, mathematics, or to indicate lesser importance, rather than physical dimensions.
A minha casa é menor do que a tua.
(My house is smaller than yours.)
O risco de perder dinheiro é menor neste investimento.
(The risk of losing money is smaller in this investment.)
O irmão dele é menor de idade.
(His brother is under age (literally: a minor).)
Três é menor que cinco.
(Three is smaller than five.)
Este problema é menor quando comparado ao outro.
(This problem is smaller (minor) when compared to the other one.)
Mais pequeno
A2This phrase literally translates to "more small". In Portugal, mais pequeno is gramatically accepted and the preferred term when describing physical size or dimensions of objects and people. In Brazil, using mais pequeno is often considered grammatically incorrect or childish, except in the specific case where one compares two different qualities of the same subject.
O teu carro é mais pequeno do que o meu.
(Your car is smaller than mine.)
Quero o pedaço mais pequeno do bolo.
(I want the smaller piece of the cake.)
Este apartamento é mais pequeno do que arejado.
(This apartment is smaller than it is airy (comparing two qualities).)
Ele parece mais pequeno nestas calças.
(He looks smaller in these pants.)
A caixa azul é a mais pequena de todas.
(The blue box is the smallest of them all.)
Summary
To summarize, use menor for almost all contexts in Brazil; using mais pequeno there sounds uneducated unless comparing two adjectives (e.g., "more small than ugly"). In Portugal, distinguish the context: use mais pequeno for physical sizes (objects, height) and menor for abstract concepts (value, age, quantity, importance).