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!}

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)