Of the currently living Nordic languages, there is far and away one standout in terms of similarity. Icelandic.

Orthographically (writing conventions), not much has changed. Consonant clusters got broken up e.g. maðr becoming maður, and unstressed fricatives (/f/) became voiced (/v/). Indeed, many modern Icelandic speakers have little trouble reading written Old Norse.

Phonetically, there have been changes. Diphthong pronunciation has varied over the past millennium. Similarly there has been a slight warping in pronunciation across the board. Simple phrases like "I am" went from ek em to ég er for example. A degree of change you might find travelling from Boston MA, to Waco, TX as the crow flies.

Another excellent sub in which to pose this question, and questions like it is /r/linguistics.