What are the differences between a UHF (as used in cryptography) and a cryptographic hash function?
The usual definition for a hash function - to avoid hard coded collisions, i.e. trivial formal adversaries have a collision as part of their description and just output it- is as a pair of algorithms $(\operatorname{Gen},\operatorname{Hash})$. In this definition $\operatorname{Gen}$ typically picks a &quot;key&quot; for the hash function, which for the &quot;keyless&quot; ones can usually be seen as the details of the description, like the exact structure and constants.
The advantage of an adversary $\mathcal A$ against the collision resistance of a scheme $H$ is then defined using the following experiment:

$k\gets \operatorname{Gen}(1^\kappa)$
$(m,m')\gets\mathcal A(1^\kappa,k)$
$\mathbf{Adv}_H^{\text{CR}}(\mathcal A;\kappa)=\Pr[\operatorname{Hash}(k,m)=\operatorname{Hash}(k,m')\land m\neq m']$

A scheme is then called collision-resistant if for all polynomial-time adversaries $\mathcal A$ the advantage $\mathbf{Adv}_H^{\text{CR}}(\mathcal A;\kappa)$ is negligible in $\kappa$.
This definition essentially amounts to &quot;give the adversary a full description of the scheme and then they win if they find a collision&quot;. The key serving the role of the definition of the hash function having only existed for so long in the real world, i.e. the key being given to the adversary the moment the hash function design was finalized.
Now for a UHF on the other hand the security definition is much weaker. The advantage of an adversary $\mathcal A$ against the universal hashing security of a scheme $H$ is then defined using the following experiment:

$(m,m')\gets\mathcal A(1^\kappa)$
$k\gets \operatorname{Gen}(1^\kappa)$
$\mathbf{Adv}_H^{\text{UHF}}(\mathcal A;\kappa)=\Pr[\operatorname{Hash}(k,m)=\operatorname{Hash}(k,m')\land m\neq m']$

A scheme is then called universal hashing secure if for all polynomial-time adversaries $\mathcal A$ the advantage $\mathbf{Adv}_H^{\text{UHF}}(\mathcal A;\kappa)$ is negligible in $\kappa$.
As you can see for the UHF security, the adversary doesn't get a full description of the scheme. They don't even get oracle access - i.e. they can't query values to get their hashes. So they have to come up with a collision that works for a non-negligible amount of instantiations (&quot;keys&quot;) of $H$. Obviously this definition is much easier to satisfy. In cryptographic practice dedicated UHF schemes actually even break down in security if an adversary learns a single input-output pair, allowing an adversary to completely recover the key $k$ and produce collisions at will.