The Conditional Logistic Regression applies fixed effects (in the context of econometrics), where each pair of subjects has an individual intercept. It can be implemented with clogit() of package survival or clogistic() of package Epi. Generalized Linear Mixed Models (GLMM) for binary data can adopt link functions like logit, probit and cloglog. We can estimate GLMM using glmer() of package lme4.

As to the choice between conditional logistic regression and GLMM for binary data, some people are in favor of conditonal (fixed-effects) logistic regression and GLMM with probit link, but against fixed-effects probit or GLMM with logit link. The reason may be that some of the consistency properties break down, especially with small within-cluster sample size (ni=2 for your case).

I am not sure wether I understand the difference between the two models. Could you please provide the reference to help clarify the confusion? Thanks