What is the difference between regular and "twisted" ECC curves? When I do:

openssl ecparam -list_curves


I get, among other entries:

brainpoolP512r1: RFC 5639 curve over a 512 bit prime field
brainpoolP512t1: RFC 5639 curve over a 512 bit prime field


Apparently the "t" means it is a twisted ECC curve. Is this slightly more secure or slightly less secure? I'd rather give up a few milliseconds of performance than give up any security margin.
TL;DR: There is no difference. 

Given an elliptic curve $E$ defined over $\mathbb{F}_p$ for some prime $p$, we say that a second curve $E_t$ defined over $\mathbb{F}_p$ is a twist of $E$ when $E_t\cong E$. That is, when there is an isomorphism between $E_t$ and $E$, defined over $\bar{\mathbb{F}}_p$ (the algebraic closure of $\mathbb{F}_p$).

From this we can conclude that every curve is a twisted curve, as every curve is isomorphic to itself. Thus the definition of a regular curve versus a twisted curve is nonsensical, there is no difference.

You may be left wondering why we care about twists in the first place. Well, it turns out that given some curve $E/\mathbb{F}_p$, in some cases one can force a user to work on some twist of $E$ instead (there could be many twists). This twist could have different security properties (notice the definition of twist with respect to $E$), which could in turn lead to an attack. This is an example of a so-called invalid-curve attack.

Edit: Note that ${\tt brainpoolPXXXr1}$ and ${\tt brainpoolPXXXt1}$ are trivial twists (see Definition 9.5.1). That means that the security properties are essentially the same. The reason why both these curves are specified, is because ${\tt brainpoolPXXXr1}$ is pseudo-randomly generated (and therefore supposedly leaving them unable to create backdoors), yet has large curve parameters. By specifying ${\tt brainpoolPXXXt1}$ which has $A=-3$, we can make some improvements in the curve arithmetic, making operations more efficient (see EFD).