They have the same general meaning, until now, but they are used in different contexts, and they speak of two very different nows.

The present time which is referred to as this day is the present era, a vague timespan which may extend for years or decades into the past and future. When we say something like “The anniversary is celebrated to this day” or “That custom continues to this day”, we don’t mean that the anniversary is being celebrated or the custom observed on the day of speaking: we mean that the celebration or observation still takes place whenever it is appropriate.

The present time which is referred to as date, however, is the specific day on which the speaker utters the phrase to date. This phrase is typically used to report the current status of some activity or expected event:

We have sold 2,500 units to date, and expect to sell 3,200 by the end of the year.
To date he has not responded to my letter.

Note that the verbforms employed with these phrases differ. To this day typically modifies a clause cast in the simple present or progressive present, reflecting an action which is repeated or habitual; to date typically modifies a clause cast in the present perfect, reflecting a current outcome of prior events.