If we look at the TLF we can see that tête means :

A. − une partie du corps.
B. − le siège de l’activité cérébrale, ou considérée du point de vue des activités intellectuelles et du psychisme.

A. would be “the head”.
B. would be “the mind”.

Tête will translate “mind” very often. Especially (but not only) in relation to thinking.

• J’ai des idées plein la tête.
• Je n’arrive pas à me sortir ça de la tête.
• Il a une tête bien faite.

Esprit will also translate “mind”, often (but not only) when considering a more global approach to the mind.

• Ça ne me vient pas à l’esprit pour l’instant.
• Avoir l’esprit logique.
• Avoir l’esprit ouvert / vif.

At times both can be used:

• Perdre la tête. / Perdre l’esprit.

Very often English phrases with mind will translate with neither tête nor esprit.

• Je pense beaucoup (plus fréquent je pense comme construction que “j’ai la tête pleine d’idées” qui se dit aussi).(My mind thinks a lot)

• À quoi penses-tu ? (What’s on your mind?)

• Je suis très préoccupé. (I have a lot on my mind.)

• La télékinésie c’est la faculté de pouvoir déplacer des objets par la pensée. (The ability to move objects with one’s mind).*

A non native will probably have to learn the French expressions in order to use the right word.

I answered strictly within the scope of your question (tête vs esprit) but I want to add that esprit has other meanings in French:

• wit (Il a de l’esprit)
• spirit (avoir l’esprit d’équipe)

^{* Merci à @Johnmacward & @Gilles for mentioning pensée.}

The Latin word, “Mens, mentis” has given a word in every Romance language except French. “La mente” in Spanish, Italian, Portuguese. “La minte” in Romanian. “La ment” in Catalan. Etc. In French, you can translate “Mind” with “l’esprit”, “la raison”, “la tête”, depending on the context. But, to me these words only translate “Mind” partially. Something is missing, some of the deeper meaning gets lost. On top of that, “l’esprit” also means “Spirit”, so “l’esprit” means both “Mind” and “Spirit” !

In a similar odd way, “Awareness”, “Consciousness”, and “Conscience” all translate into, “La conscience” in French !

Interestingly enough, the latin saying “mens sana in corpore sano” is famously known in France as “Un esprit sain dans un corps sain”.

Not a reply as Laure already wrote a good one, but a note about mēns heritage.

La mente (and sometimes le ment), meaning intelligence, entendement, décision, used to exist in French.

Its usage possibly dropped because of multiple collisions with :

• la mente meaning “lie” (now mensonge), from mentiō

• la ment(h)e meaning “mint”, from mentha..

• la mante, a kind of cape/coat (relates to manteau), from mantellum

• la mante (religieuse), from mantis.

• l’amante, the loved/loving woman, from amāns.

Note that there are words sharing the same root like dément and mental and the adverbial suffix -ment is common. There is also mention and mentir.