I would like an explanation for the term "connectivity" in the question "Do the compounds have the same connectivity? I do not understand what "connectivity", a vague term, means.
Connectivity describes how the atoms are connected to each other. Consider a daisy chain (image courtesy of Wikipedia).
All the daisies are similar enough much like carbon atoms are similar enough between them; we cannot distinguish. Counting ten daisies in that chain, I could also represent it as (with the ends of the chain reconnecting):
This compound would by cyclodecane.
Now imagine you replaced two daisies with violets. Of course, violets are immediately distinguishable from daisies like oxygen atoms are from carbon atoms. We now have a number of different options how to replace daisies with violets. They could be next to each other, separated by one, two three or four daisies. That would give rise to the following structures:
In connectivity terms, in the first case and the first case only the two violets are connected to each other. In the second case and the second case only, there is a single daisy connected to two violets. These should be the obvious cases of different connectivities, although each different violet placement gives rise to a new connectivity.
These compounds are different; especially the first (which is a peroxide) and the second (which is the formyl acetal of a dialcohol) have very different chemical (and physical) properties from the others (which are simply cyclic diethers).
Added difficulty arises for issues that I have purposefully ignored in my daisy chain analogy: it does not consider three-dimensionality. Of course, atoms in reality are connected in a three-dimensional manner. You want to be sure that the connectivity is the same in all three dimensions (but not necessarily in the same orientations). Taking butan-2-ol as an example, we speak about identical connectivity whether we are dealing with (R) or (S)-butan-2-ol; because the same carbons atom are still connected to the same other atoms (e.g. carbon 2: Connected to a methyl group, an ethyl group, a hydrogen and a hydroxy group). But a different connectivity would be butan-1-ol, where carbon 2 is suddenly connected to a hydroxymethyl group and a second hydrogen (in addition to the ethyl group and the hydrogen which remained unchanged).