<p>I would say that Penn and Columbia are considered more selective than UCB for California students. I would say that for prestige, roughly, UCB = JHU,Emory, or Wash U at least based on East Coast perception.
Columbia is about as hard to get into as Stanford so it’s not really comparable.
It seems to me that UCB is in the top few world-caliber public universities with UMich, UNC-CH, and UVA, with the next group far below.
The undergraduate experience is likely to be more pleasant at private universities simply because they have more money but that doesn’t mean the students aren’t as strong or the faculty less gifted researchers. For engineering, I agree that they’re even in terms of selectivity, with an edge to UCB. On the East Coast, UCB is not considered superior to Columbia – it really depends. Nation-wide, their name recognition is the same, overall, considering they’re not football/basketball powerhouses and the people who need to know about both know enough not to make a difference.</p>
<p>OP: I would say matches for you would be NYU, Emory, Tufts, Brandeis, Carnegie Mellon (not CS), perhaps Cornell.
A university with 20% or less admit rate is automatically a crapshoot so JHU would be a match if not for selectivity. It’s difficult for high-stats applicants to find true matches (UCB would be one, because you’re in-state).</p>