As far as I can understand, you are asking two separate questions.
First, you ask if using but instead of yet as a conjunction is 'stronger'. The answer is no. There is no inherent 'strength' in either of these words. They are interchangeable when used as conjunctions. Both can be used in ways in which they are not interchangeable ('none but the brave', 'he hasn't finished yet'), but in common use they are synonyms.
Second, you ask if although or nevertheless was correct. In the context of the sentence you provided, nevertheless is correct. Nevertheless is an adverb; although is a conjunction. You could have used although, but you would, in effect, have been creating one very long run-on sentence, regardless of whether you used a full-stop/period before it or not. Nevertheless was the better choice as it meant you created a proper new sentence.
It is quite difficult to spot that nevertheless is an adverb, above all because it does not end in the usual '-ly', but it does in fact modify the verb 'role' in your sentence. Consider:
Technology has a role, nevertheless, in children's education.
You can see more clearly here that 'nevertheless' relates to the verb 'role'. It is not joining two related but separate parts of sentence like a conjunction does. You could take out 'nevertheless' and you'd have one simple sentence.
With a conjunction, you need to have the 'other part' of the sentence that it is linking to.
Although technology has a role in children's education, ... [the other part of the sentence is required here for it to make sense].
Furthermore, while it is not accurate to say (as many pedants used to) that 'you can't start a sentence with a conjunction', a sentence with a conjunction does have to link to another clause or immediately preceding sentence in an obvious way. For the purposes of an English language learner, one might be better to pretend that the old 'rule' still holds. Avoid starting sentences with conjunctions unless you're sure you know what you're doing.