It’s not really the same meaning : “date limite” is a milestone, “délai” is a duration. But you can say : “Je t’accorde un délai jusqu’à cette date limite”.

These two words are not synonyms. Date limite refers to a point on the time line and délai to a period of time.

Quel est la date limite pour rendre ce travail ?
Tu dois avoir fini pour vendredi soir/ la fin de la semaine.

Quel est le délai pour faire ce travail ?
Tu as dix jours.

It seems to me that date limite is closer to “deadline” than to délai, although date limite does not always translated as “deadline”. For example, date limite de vente is “sell by date”.

I can’t think of an instance where I’d translate délai a “deadline”.

• Quel est le délai pour rendre ce travail ? → How long have I got to hand in this work?
• Le délai de livraison est de 3 jours. → Allow 3 days for delivery.
• Le délai d’attente moyen est d’au moins huit jours. → The average waiting time is at least a week.

Sometimes délai is translated as “delay”, but I can only think of one instance.
Puis-je avoir un délai pour finir ce travail ? →
Can I have a delay to finish that work?

“Data limite” would be deadline/milestone, whereas “delai” is more like leadtime/delay, as in “delay line” in electronics, precisely. For sure, “delai” refers to a duration, whereas “date limite” refers to a point in time.