I think the entries on SEP can answer your question better than I can:
“Hegel’s dialectics” refers to the particular dialectical method of argument employed by the 19th Century German philosopher, G.W.F. Hegel (see entry on Hegel), which, like other “dialectical” methods, relies on a contradictory process between opposing sides. Whereas Plato’s “opposing sides” were people (Socrates and his interlocutors), however, what the “opposing sides” are in Hegel’s work depends on the subject matter he discusses.
The term Temporal Logic has been broadly used to cover all approaches to reasoning about time and temporal information, as well as their formal representation, within a logical framework, and also more narrowly to refer specifically to the modal-logic type of approach introduced around 1960 by Arthur Prior under the name Tense Logic and subsequently developed further by many logicians and computer scientists.
The differences here are that Dialectical logic is principally concerned with analysis via appraisal of contradictory processes between opposing sides. Temporal or tense logic is logic about time. For example, "no event precedes itself". Tense logic is a formal logic about time.
https://plato.stanford.edu/entries/hegel-dialectics/
https://plato.stanford.edu/entries/logic-temporal/