Large cardinals without replacement
There is a simple example of a large cardinal axiom that extends $\sf ZC$ and is stronger than $\sf ZFC$ and yet it can violate $\sf ZFC$. Take for example the theory with the following axioms:
Universes: every set belongs to a Grothendieck universe.
Denumerability: No Grothendieck universe can have more than finitely many Grothendieck universes inside it
Separation: as in $\sf ZC$.
Where a Grothendieck universe can be defined as an extensional well founded transitive set that is closed under powerset, set unions and not-greater in cardinality than operators.
This theory clearly violates $\sf ZFC$, yet it does interpret $\sf ZFC$, and even some versions of $\sf MK$
By the way the wholeness axiom can be considered as an extension of $\sf ZC$, since it doesn't really extend $\sf ZFC$ for all formulas of its language!
Overall, the large cardinal axiom hierarchy is very similar between ZC (ZFC minus replacement; we are including regularity) and ZFC. A large cardinal axiom (unprovable in ZFC) satisfied by $κ$ typically implies in ZC that $V_κ$ satisfies ZFC + weaker large cardinal axioms. However, the axioms typically do not imply additional replacement above $κ$ even for $Σ^V_3$ axioms such as existence of a strong or a supercompact cardinal (and hence their strength is truncated accordingly); but an extendible or a proper class of strong cardinals gives $Σ_2$ replacement, and a proper class of extendibles gives $Σ_3$ replacement.
Equiconsistencies also typically carry over to ZC. For example, ZC + $L(ℝ)⊨\text{AD}$ is equiconsistent with ZC + $ω$ Woodin cardinals whose supremum exists.
However, there are some differences.
One is notational. ZC does not prove that $ω2=ω+ω$ exists as the transitive set. However, ZC interprets $Σ_1$ replacement, and we can either add $Σ_1$-replacement, or implicitly speak of codes for ordinals and other sets.
Defining HOD requires replacement. However, there are inequivalent first order definable approximations of OD in ZC, one of which is $∪_{V_κ \text{ exists}} \mathrm{OD}^{V_κ}$.
The lack of singular infinite cardinals creates some strength differences. For example, ZC + $∀κ \, (κ^+)^L < κ^+$ is equiconsistent with ZC rather than ZC + $0^\#$. Still, there are other covering properties whose violation has high strength in ZC, and nonexistence of an inner model with a Woodin cardinal should still imply that the core model exists. Also, ZC($j$) + “$j$ is a nontrivial elementary embedding $V→V$” (called Wholeness Axiom; it proves ZFC without replacement for $j$-formulas) is consistent relative to the $\mathrm{I}_3$ axiom, as opposed to the Kunen inconsistency in ZFC($j$) due to the axiom of choice and existence of $j^ω(\mathrm{crit}(j))$.
There are large cardinal axioms for ZC that are implied by ZFC. Borel determinacy is equivalent to $∀r∈ℝ \,∀α < ω_1 \, ∃β \, L_β(r) ⊨ \text{“} ω_α \text{ exists”}$. Also, the least $α<β$ with $\mathrm{Theory}(V_α)=\mathrm{Theory}(V_β)$ are strictly between $ω_1^L$ and $c^+$.
Bounded quantifier ZC (BZC, also called Mac Lane set theory) is useful for some equiconsistency statements, and as a minimal base theory. For example, BZC + a proper class of Woodin cardinals is conservative over $\text{Z}_2 + \text{PD}$ (second order arithmetic with projective determinacy). In turn, key theorems about universally Baire sets relying on a proper class of Woodin cardinals can in fact be proved in BZC + a proper class of Woodin cardinals (even though it does not prove that the set of all universally Baire sets of reals exists).
By reflection, for every consistent axiom A, ZC+A has lower consistency strength than ZFC+A, but this need not apply to schemas. For example, Vopěnka's principle over ZC (or just BZC) implies ZFC.