There are two differences here, case and number. Let's take a look at some examples.
Nominative Case
Ein Tag wie dieser (Tag).
(A day like this(or that) day)
(Sg+Sg)
(You're talking about a specific day, compared with another specific day.)
Ein Tag wie diese (Tage).
(A day like these days)
(Sg+Pl)
(You're talking about a specific day, compared with a common feature of several other days.)
Tage wie dieser (Tag).
(Days like this one)
(Pl+Sg)
(You're talking about several days, all similar in one feature to a certain other day.)
Tage wie diese (Tage).
(Days like these days)
(Pl+Pl)
(You're talking about several days that all share a particular feature or features with some other days.)
Genitive Case
Wie schön war doch der Morgen des Tages wie dieser.
(Oh how beautiful was the morning of a day like this one)
(Sg+Sg)
Wie schön war doch der Morgen des Tages wie diese.
(Oh how beautiful was the morning of a day like these days)
(Sg+Pl)
Wie schön war doch der Morgen der Tage wie dieser.
(Oh how beautiful was the morning of days like this one)
(Pl+Sg)
Wie schön war doch der Morgen der Tage wie diese.
(Oh how beautiful was the morning of days like these)
(Pl+Pl)
Dative Case
Er geht einem Tag wie diesem aus dem Weg.
(He avoids a day like this one) (=He does his best to avoid such situations.)
(Sg+Sg)
Er geht einem Tag wie diesen aus dem Weg.
(He avoids a day like these days)
(Sg+Pl)
Er geht Tagen wie diesem aus dem Weg.
(He avoids days like this one)
(Pl+Sg)
Er geht Tagen wie diesen aus dem Weg.
(He avoids days like these days)
(Pl+Pl)
Accusative Case
Ich wünsche mir einen Tag wie diesen.
(I am hoping for a day like this one)
(Sg+Sg)
Ich wünsche mir einen Tag wie diese.
(I am hoping for a day like these days)
(Sg+Pl)
Ich wünsche mir Tage wie diesen.
(I am hoping for days like this one)
(Pl+Sg)
Ich wünsche mir Tage wie diese.
(I am hoping for days like these days)
(Pl+Pl)