The first adverb modifies the second one, so the two sentences have slightly different meanings.

Mieux comprendre = to better understand

On comprend mieux = one understands better

On comprend tout de suite mieux = right away, one understands better

On the other hand:

Comprendre tout de suite = to understand right away

On comprend tout de suite = one understands right away

On comprend mieux tout de suite = one understands better right away [than at another time]

With this sentence, "tout de suite mieux" is probably the one that makes sense, but here’s another sentence in which the opposite would work:

Dépêche-toi, ça se mange mieux tout de suite.

Hurry up, it’s better when you eat it right away.

Note putting the second adverb between commas can change the meaning. In such a situation the adverb might modify the entire sentence:

On comprend mieux, tout de suite, pourquoi c’est arrivé.

= Tout de suite, on comprend mieux pourquoi c’est arrivé.