Offrir is to give as a present. In your sentence, what’s understood is that the silver bracelet is a gift. Donner is has a broader meaning, offrir a more specific one.

• Ma mère m’a offert un bracelet en argent pour mon anniversaire.

could be translated as :

• My mother gave me a silver bracelet as a birthday present.

Offrir is to give someone a present.

• Mes parents m’ont offert un stylo pour mon anniversaire.

It can also be used for non material things, for example :

• Il m’a offert son amitié.¹

• Il m’a offert son aide.²

In those two preceding sentences offrir could be replaced by proposer, offrir there implies the recipient may refuse whatever (help or friendship or other) they are being offered.

Donner is to give because you want this person to have something, you decide to do it whether the recipient wants it or not.

• Je donne mon sang régulièrement.³
• J’ai donné un de mes stylos à mon copain parce qu’il avait perdu le sien. ⁴

Theoretically you can refuse what is being offert, refusing what someone donne is more difficult, it can be considered as an offence.

But the limit between the two is sometimes very thin and a lot of people use donner instead of offrir and you might hear :

• Mes parents m’ont donné un stylo pour mon anniversaire.

Here an excerpt from a law firm’s website:

Offrir ouvre liberté au bénéficiaire d’accepter ou pas, induit l’idée d’une mise à disposition, sans lien nécessaire avec la gratuité, plus prégnante dans l’acte de donner, sauf lorsqu’il est synonyme de remettre ou confier.

Donner éloigne le refus, qui serait perçu comme un sacrilège ou une discourtoisie. La donation a d’ailleurs une résonance juridique puisque emporte changement de propriétaire et mutation gracieuse dans les deux sens de cette souriante épithète.

¹ He offered me his friendship.
² He offered me his help.
³ I give blood regularly .
⁴ I gave my mate one of my pens because he’d lost his.

I

Every time you can say “make a gift of sth to sb” or confer gratuitously to sb the enjoyment of some amenity you can translate by “offrir qqc à qqn“. In the case of amenities “offrir” can be replaced by “proposer”.

II

When the translation explained in “I” is valid the only difference in the two verbs is one of formality, in the official sense of the word for some cases and in its social sense for others; it is always possible to replace “offrir” by “donner” but then you make abstraction of a particular context of conventions of gift making or of host/guest conventions. This is true for another verb that means basically “to give” : “léguer”.

• Sa mère lui a offert un ordinateur pour son quinzième anniversaire.
• Sa mère lui a donné un ordinateur pour ses quinze ans. (Abstraction is made of any particular convention. It might have been the case that the mother thought it was high time her daughter had a computer, with no idea of making a conventionnal gift, as possibly she never did that.)

On the count of formality, a difference to keep in mind is that “offrir” has two closely connected meanings; one is that of proposing to someone to accept a gift and the other is that of having proposed a gift, having had the proposition accepted and finally of having made that gift. (Under French law important offers of property have to be accepted in order to proceed to their transfer.) This is not the case for “donner”, wherefrom one understands better the lack of formality associated with “donner”.

• Elle a légué sa fortune à une université, le legs devenant effectif après sa mort.
• Elle a offert sa fortune à une université, le don prenant effet après sa mort. (not common)
• Ella a donné toute sa fortune à une université, le don prenant effet après sa mort.
  (Here, one makes abstraction of the legal aspect of the act of giving away something.)

III

Evidently, “donner” is in no way replaceable when is missing a context of conventions for gift giving and so on. A few examples will impart certain essential ideas.

• Il a donné de l’argent à son fils pour qu’il aille acheter les provisions. (Obviously there can’t be a context of special conventions; here, by “donner” one means simply “to hand out sth”; the same thing is true for similar verbs, such as “pass on” for instance.)

Here are a couple of examples showing that a given act, in its most basic nature, can belong to either of the contexts (conventions/no conventions).

• Une hôtesse offrit un verre de jus de fruit aux visiteurs. (visit in a fruit juice factory; the visitors are guests.)
• La mère donna/tendit/offrit un verre de jus de fruit à son fils. (mother having meals with her children; a young mother with small chidren: “donna/tendit”. An older woman having married sons she sees only from time to time: (donna/tendit)/offrit, according to the particular distance between them.)

“offrir” has the (positive) meaning of giving a gift, something valuable.

“donner” can be negative, for exemple “ma mère m’a donné une gifle”: “my mother gave me a slap”.

Depending on being the one who gives or who receives, the verb might be different, too:

1. “When I was starved/wounded, some people offered me a meal/his blood”:

   • I am grateful for the gift, so “offrir” (to offer) is the right answer.
   • Say “donner” (to give) here would mean I am ungrateful.

2. “When he was starved/wounded, I gave him a meal/my blood”:

   • As the one who makes the gift, I am not supposed to highlight me, so I use the least significant verb (donner/to give) rather than the most significant one (offrir/to offer).

Does it help?