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As for your example, It makes me think of home.

It sounds like you have a lucid image of hometown, therefore you can say 故郷が思い浮かぶ.

On the other hand, 故郷が浮かぶ sounds like images just simply pop up in your head and it's not so important or you have a vague image of hometown. Since 浮かぶ connotates with an expression like 雑念が浮かぶ: miscellaneous thought comes into mind. which are unstable images. Therefore, in my opinion, 故郷が思い浮かぶ sounds you are retrospecting your hometown rather than saying 故郷が浮かぶ.

As for your example, It makes me think of home.

It sounds like you have a lucid image of hometown, therefore you can say 故郷が思い浮かぶ.

On the other hand, 故郷が浮かぶ sounds like images just simply pop up in your head and it's not so important or you have a vague image of hometown. Since 浮かぶ is associated with an expression like 雑念が浮かぶ: miscellaneous thought comes into mind. which is a collection of unstable images. Therefore, in my opinion, 故郷が思い浮かぶ sounds you are retrospecting your hometown rather than saying 故郷が浮かぶ.

As for your example, It makes me think of home. It sounds like you have a lucid image of hometown, therefore you can say 故郷が思い浮かぶ. Similarly but, 浮かぶ usually refers to an instantaneous image, for example, If you forgot something at your home and you don't know where it is, but anyway you rush back home. You can say 忘れ物が浮かんだから家に帰る。 I think I forgot something at home. Anyway I go back home. My analysis of The difference between 浮かぶ and 思い浮かぶ in the head not for outer things like facial expression. 思い浮かぶ normally requires conscious effort you have made some attempt for it, "浮かぶ" normally doesn't require conscious effort, simply it is just popping up.

I think 何かが浮かぶ　is more of a flash thought. It just contains not necessarily so clear mental image such as "Oh, I was boiling water.I forgot it.I need to turn it off" It is more of instantaneous thought. I mean it is used for miscellaneous thought in your head.

On the other hand, I think 何かが思い浮かぶ is more of a clear thought, better resolution of the mental image. And it tends to associate with one particular concept, not like miscellaneous, vague images. As you pick up "It makes me think of home" since you can remember what your home or hometown in detail. Anyway, I think in most case it's used as a transitive verb.

Example usage: it might be not so useful anymore. Since you already have another answer

「してやったり！」の表情を浮かべる:

His face shows "That's what I expected!"

不敵な笑みを浮かべる.

The dog's face shows fearless smile

I think you can use "浮かぶ”　for more spontaneous/natural things, which nor necessarily requires so much effort.

彼女の顔から自然な笑みが浮かぶ。

Her face shows natural smile.

You can use it for holding an idea.

良いアイデアが浮かんだ。

I have hit on a good idea.

I think 思い浮かべる/思い浮かぶ is more of making a conscious effort.

鍵をどこに置いたか思い浮かべる。

I am trying to recall where I put my key.

昔、旅行した場所の風景を思い浮かべる。

I am remembering the scenery where I traveled before.

Bonus: It's relevant, not necessary.

チェシャ猫の様にニヤニヤ笑いを浮かべる。

grinning like a Cheshire cat

On the other hand, チェシャ猫抜きのニヤニヤ笑いが浮かぶ。. Curious case.

a grin without Cheshire cat

Is it a facial expression without a face?

As for your example, It makes me think of home.

It sounds like you have a lucid image of hometown, therefore you can say 故郷が思い浮かぶ.

On the other hand, 故郷が浮かぶ sounds like images just simply pop up in your head and it's not so important or you have a vague image of hometown. Since 浮かぶ connotates with an expression like 雑念が浮かぶ: miscellaneous thought comes into mind. which are unstable images. Therefore, in my opinion, 故郷が思い浮かぶ sounds you are retrospecting your hometown rather than saying 故郷が浮かぶ.

As for your example, It makes me think of home.

If you have a clear/detail image of hometown, probably you can say 故郷が思い浮かぶ.

Probably having an image for a short period of time, like the following case: If you forgot something at your home and you don't know where it is, but anyway you rush back home.

忘れ物が浮かんだから家に帰る。

I think I forgot something at home. Anyway I go back home.

My analysis of The difference between 浮かぶ and 思い浮かぶ in the head not for outer things like facial expression. 思い浮かぶ normally requires conscious effort you have made some attempt for it, "浮かぶ" normally doesn't require conscious effort, simply it is just popping up.

I think 何かが浮かぶ　is more of a flash thought. It just contains not necessarily so clear mental image such as "Oh, I was boiling water.I forgot it.I need to turn it off" It is more of instantaneous thought. I mean it is used for miscellaneous thought in your head.

On the other hand, I think 何かが思い浮かぶ is more of a clear thought, better resolution of the mental image. And it tends to associate with one particular concept, not like miscellaneous, vague images. As you pick up "It makes me think of home" since you can remember what your home or hometown in detail.

Anyway, I think in most case it's used as a transitive verb.

「してやったり！」の表情を浮かべる:

His face shows "That's what I expected!"

不敵な笑みを浮かべる.

The dog's face shows fearless smile

I think you can use "浮かぶ”　for more spontaneous/natural things, which nor necessarily requires so much effort.

彼女の顔から自然な笑みが浮かぶ。

Her face shows natural smile.

You can use it for holding an idea.

良いアイデアが浮かんだ。

I have hit on a good idea.

I think 思い浮かべる/思い浮かぶ is more of making a conscious effort.

鍵をどこに置いたか思い浮かべる。

I am trying to recall where I put my key.

昔、旅行した場所の風景を思い浮かべる。

I am remembering the scenery where I traveled before.

チェシャ猫の様にニヤニヤ笑いを浮かべる。

grinning like a Cheshire cat

On the other hand, チェシャ猫抜きのニヤニヤ笑いが浮かぶ。. Curious case.

a grin without Cheshire cat

Is it a facial expression without a face?

As for your example, It makes me think of home. It sounds like you have a lucid image of hometown, therefore you can say 故郷が思い浮かぶ. Similarly but, 浮かぶ usually refers to an instantaneous image, for example, If you forgot something at your home and you don't know where it is, but anyway you rush back home. You can say 忘れ物が浮かんだから家に帰る。 I think I forgot something at home. Anyway I go back home. My analysis of The difference between 浮かぶ and 思い浮かぶ in the head not for outer things like facial expression. 思い浮かぶ normally requires conscious effort you have made some attempt for it, "浮かぶ" normally doesn't require conscious effort, simply it is just popping up.

I think 何かが浮かぶ　is more of a flash thought. It just contains not necessarily so clear mental image such as "Oh, I was boiling water.I forgot it.I need to turn it off" It is more of instantaneous thought. I mean it is used for miscellaneous thought in your head.

On the other hand, I think 何かが思い浮かぶ is more of a clear thought, better resolution of the mental image. And it tends to associate with one particular concept, not like miscellaneous, vague images. As you pick up "It makes me think of home" since you can remember what your home or hometown in detail. Anyway, I think in most case it's used as a transitive verb.

Example usage: it might be not so useful anymore. Since you already have another answer

「してやったり！」の表情を浮かべる:

His face shows "That's what I expected!"

不敵な笑みを浮かべる.

The dog's face shows fearless smile

I think you can use "浮かぶ”　for more spontaneous/natural things, which nor necessarily requires so much effort.

彼女の顔から自然な笑みが浮かぶ。

Her face shows natural smile.

You can use it for holding an idea.

良いアイデアが浮かんだ。

I have hit on a good idea.

I think 思い浮かべる/思い浮かぶ is more of making a conscious effort.

鍵をどこに置いたか思い浮かべる。

I am trying to recall where I put my key.

昔、旅行した場所の風景を思い浮かべる。

I am remembering the scenery where I traveled before.

Bonus: It's relevant, not necessary.

チェシャ猫の様にニヤニヤ笑いを浮かべる。

grinning like a Cheshire cat

On the other hand, チェシャ猫抜きのニヤニヤ笑いが浮かぶ。. Curious case.

a grin without Cheshire cat

Is it a facial expression without a face?