He gained entry into the building.
He gained access into the building.
Any difference in usage or meaning?
In Modern English, according to the OED, access is defined, in this context, as "The power, opportunity, permission, or right to come near or into contact with someone or something; admittance; admission."
Entry, on the other hand, is defined as "The action of coming or going in," in this context (while the word does have a definition listed which is synonymous with access, that definition is considered obsolete).
In other words, to gain entry into the building means to get into the building. This can be done sneakily, or with permission, or even by brute force, barrelling your way in.
On the other hand, to gain access literally means that you have gained the power or "permission" to enter the building. You are being allowed into the building by the building's security (whether this security is a system of guards or a computer system), or you can reasonably get past the guard.
Obviously, this can be done in a number of ways, from hacking into a computer mainframe to access security keys, to pickpocketing a guard for literal keys, to dressing up and presenting yourself as a high-authority figure, to buying an admission ticket. However, at this point in time, if you have "gained access" to the building, you now have what can be essentially described as an unlocked and unblocked pathway into the building.
If you get in, you have gained entry.
If you are allowed in, you have found a way you can get in easily, or if you have gotten in and you pose little risk of being removed, you have gained access.
In your example, I don't see any real difference in meaning. Perhaps there is a small implied difference:
"Access" - he acquired access through the proper (legal) means.
"Entry" - holds (for me) a slight implication that maybe he didn't use "legal" or "proper" methods.
Generally speaking, "access" means being able to make use of, approach, enter or exit (see here), whilst "entry" is the act of entering, and perhaps more importantly, the privilege or right of entering. See here.
However, since you specify "into the building", in my opinion, the difference is nullified. That said, your second sentence sounds better as "to" rather than "into" the building, since "enter/entry" already carries the connotation of "in".