The vines created by Entangle are flammable because regular vines are flammable

There are no secret rules.

D&D is written using natural language. Unless the game defines a particular term differently we are expected to use the natural language meaning of the word.

It uses natural language so we can use our own experience and understanding of the words in our world as a baseline, with the game providing exceptions to this understanding (like the existence of magic and monsters).

The Player's Handbook starts off the spellcasting section with this statement:

A spell is a discrete magical effect, a single shaping of the magical energies [...] into a specific, limited expression.

In other words, spells produce a limited magical effect. That limited magical effect is specific, and is described precisely by the spell description.

The Entangle spell creates:

Grasping weeds and vines sprout from the ground in a 20-foot square starting from a point within range.

The game (and the spell) do not make any further definition of weeds and vines, and thus we must use the natural language definition of weeds and vines.

Both weeds and vines can be cut, and thus slashing damage at a minimum should be able to damage the vines and thus be used to free oneself.

Similarly, being compromised of plant matter, weeds and vines are both flammable (see real word forest fires), and thus a well placed fireball should set them on fire (but that would also likely damage the creatures in the Entangle spell the fireball was attempting to free).

But grease isn't flammable...

Under the natural language meaning of grease, the substances which make up grease in our word are not universally flammable. As a result the grease created by the grease spell is not flammable by default.

There is, however leeway, for a DM, to rule that grease created by the Grease spell in their world is flammable, and still be following RAW.

Isn't this just another "secret rule"?

No.

Let us consider the inverse situation:

The vines are not flammable and are impervious to all damage because the spell doesn't say they can be damaged.

This interpretation relies on the secret rule that things aren't flammable or capable of being damaged unless they say they are.

Let's take this interpretation to the extreme. Let's say I have a spell that creates dry sticks, Create Dry Sticks. This full text of the spell is:

A cubic foot of dry sticks are created at your feet.

As the spell does not say that the sticks are flammable, they aren't flammable. Indeed, it doesn't say they can be damaged in any way.

As a result, we can't use these sticks to make a fire with. Indeed...we have created indestructible sticks! Bundle them together and we now have an indestructible weapon (made of dry sticks), or perfectly resistant armour (made of dry sticks).

You can't destroy them, unless the rules say you can

D&D spells work as stated in their description. If the rules say you can do something, you can do it. If they don't say you can do it, it becomes up for DM ruling.

As stated in an unofficial tweet by Jeremy Crawford:

Beware of claims that a rule does something mentioned nowhere in that rule or elsewhere in the core books. There aren't secret rules.

Some spells that conjure vines do tell you how you can get free, such as Entangle:

A creature restrained by the plants can use its action to make a Strength check against your spell save DC. On a success, it frees itself.

That is the only way listed to get free, so it is the only way to break free; you can't, by the rules as written, decide to try and cut or burn your way free instead. How you and your DM justify why this isn't possible, or if you want to houserule it being possible, is entirely up to you.

Perhaps the plants will just instantly regrow, or perhaps they're made of magical unobtainium that is impervious to all damage. The rules don't care, they just care about what works for game balance.

Balance concerns:

Sure, this might mess with your internal sense of logic, because vines should be destroyable/burnable, but ruling that will make the spells a lot weaker than they already are. Why bother trying to use an action to break free against the opponent's saving throw if you can just cut your way out in the same action? It will essentially mean that the spell no longer scales in any way with the casters saving throw, because you've provided a new, easy way to get out.

But vines are flammable!

People are mentioning that vines should be flammable, but if they were, the spell would have told you so, such as the Web spell:

The webs are flammable. Any 5-foot cube of webs exposed to fire burns away in 1 round, dealing 2d4 fire damage to any creature that starts its turn in the fire.

Per Jeremy Crawford:

If the grease spell created a flammable substance, the spell would say so. It doesn't say so. #DnD

If the vines were flammable or destructible, they would say so by the same reasoning. Spells that summon objects that are destructible will generally say so:

For example: Bigby's Hand:

The hand is an object that has AC 20 and hit points equal to your hit point maximum. If it drops to 0 hit points, the spell ends. It has a Strength of 26 (+8) and a Dexterity of 10 (+0). The hand doesn’t fill its space.

Besides, not all vines are flammable. In the jungle, one of the best ways of getting fresh drinking water is by cutting open vines, that are full of delicious water. You won't likely be setting those on fire any time soon.