Is Neil a straight up villain or anti-hero?
Yes, Neil is technically a criminal and somebody that we shouldn't admire, but Heat is one of the rare movies, where I actually feel sympathetic for the supposed "bad guy" and actually want to root for him. I always thought that Waingro was the true "villain" (i.e. pure evil and rotten to the core) of Heat.
This is how TV Tropes described Neil:
https://tvtropes.org/pmwiki/pmwiki.php/Film/Heat
Anti-Villain: Neil is a Type I with some shades of a Type IV. He's a thief and he's not above killing, but he's a professional with standards, who avoids violence unless necessary and is loyal to his crew. He's also depicted as a somewhat pitiable figure as his mentality of never getting too attached to anything means he lives a very lonely life.
Even Evil Has Standards: Neil is willing to commit murder if he thinks it's necessary, but usually tries to avoid that necessity. Of the three heist attempts we see, one is after midnight at a business with no night shift, while the other two are carefully choreographed plans that rely on intimidation, surprise, and nonlethal violence (with the clear threat of lethal escalation if someone doesn't cooperate) to cow security guards and witnesses. As Hanna points out, the third armored truck guard was murdered because at that point, a third murder would cost them less than a live witness to the first two, and Neil is pissed about those first two. So pissed, in fact, that he and his crew try to kill Waingro for it.
Gentleman Thief: A Deconstructed Character Archetype. Neil has the charm and all the connections, but he's painfully lonely, and won't get close to anyone for fear that the cops will be right around the corner.
Karmic Thief: Neil and his crew only target high-value targets like precious metal depositories, banks and corporate money vans. Invoked during the bank robbery scene when Neil says, "We want to hurt nobody. We're here for the bank's money, not your money. Your money's insured by the federal government, you're not going to lose a dime. Think of your families, don't risk your life, don't try to be a hero."
Pragmatic Villainy: Neil and his crew are milder cases but still, they don't seem noticeably upset about shooting the guards during the armored car robbery. And during the bank shootout, they don't show much concern for the many innocent bystanders they're endangering (when we see Neil shoot a burst at Hanna in the parking lot, one officer gets shot, and it looks like at least one bystander also gets hit as well, possibly fatally). But they're disgusted with Waingro because he escalated the armored car robbery into a bloodbath and his reckless actions are a liability to them.
https://moviechat.org/tt0113277/Heat/58c7703d93cef4080d785c9f/Sympathy-for-Neil-and-his-crew share