Grand-père vs. aïeul
While both grand-père and aïeul can be translated as grandfather, they are not interchangeable. Grand-père is the common, everyday term, whereas aïeul is a more formal, literary, and broader term that often signifies a more distant ancestor.
Grand-père
A1This is the standard and most common word for grandfather. It specifically refers to the father of one's father or mother and is used in all contexts, from casual family conversations to formal documents.
Mon grand-père aime beaucoup lire le journal le matin.
(My grandfather really likes to read the newspaper in the morning.)
Nous allons rendre visite à mon grand-père ce week-end.
(We are going to visit my grandfather this weekend.)
Quand j'étais petit, mon grand-père me construisait des cabanes dans le jardin.
(When I was little, my grandfather used to build me treehouses in the garden.)
Son grand-père maternel est né en Espagne.
(His maternal grandfather was born in Spain.)
Aïeul
C1This is a formal and literary term. While it can technically mean grandfather, it is most often used to mean ancestor or forefather, referring to a male ascendant from a generation before one's grandparents. Its plural form, aïeux, almost exclusively means ancestors. Using aïeul for a living grandfather is rare and would sound very dated or poetic.
Le portrait de son aïeul est accroché dans le grand salon.
(The portrait of his ancestor is hanging in the large living room.)
Mon aïeul a combattu durant la Première Guerre mondiale.
(My ancestor fought during the First World War.)
La fortune de la famille a été fondée par un aïeul visionnaire au 19ème siècle.
(The family's fortune was founded by a visionary forefather in the 19th century.)
Il est fier de descendre d'un aïeul qui était un grand artiste.
(He is proud to be descended from an ancestor who was a great artist.)
Le respect des aïeux est une tradition importante.
(Respect for the ancestors is an important tradition.)
Summary
In short, use grand-père in all everyday situations to talk about your direct grandfather. Use aïeul when you want to sound formal or literary, or more commonly, when referring to a more distant male ancestor or forefather. Think of grand-père as grandpa and aïeul as forebear or ancestor.