I was told they both have the same meaning "following" such as "following the previous discussion...". What is the difference in nuance between "suite à" and "à la suite de"? In which situation would they be used differently?
I would say that the main difference is in the causality link
Suite à" has a strong causality link with the event (as a consequence of)
A la suite de" is closer to its litteral translation (following) without insisting on the causality, but more on the sequence.
As a remark, "suite" can have different meanings (a suite in a hotel, a musique, an escort). For instance, you could say:
Il appartient à la suite du prince. (he is part of the escort of the prince) Il était sensible à la suite de Bach. (he liked the suite of Bach)
There certainly is a lot of overlap and they can probably be used synonymously in most places.
Where they diverge is that « suite à » has connotations of being in response to or of following on the heels of something — being caused by it.
On the other hand, « à la suite de » suggests adding to an existing sequence, further expanding it.
So you might say « suite à votre lettre » but not « à la suite de votre lettre ».
And it might be more natural to say « à la suite d'une série de promotions, Amal était devenu le chef de l'entreprise ».
This is a determination by a non-native speaker who finds vocabulary subtleties one of the harder areas to become fluent in, so I invite comments!