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Verbs that follow prefer, love, like, or dislike can take either the gerund or the infinite.
For more information, see the ELU question:When should a verb be followed by a gerund instead of an infinitive?
Hence in the case of prefer either form is acceptable :

I prefer studying
I prefer to study

To avoid repeating the same noun twice in the same sentence use the pronoun, one.

one and ones

We use one (singular) and ones (plural) to avoid unnecessary repetition.

• See those two girls? Helen is the tall one and Jane is the short one.
• Which is your car, the red one or the blue one?

• My trousers are torn.

I need some new ones.

Therefore I would advise the OP to write

I would prefer to participate in the student mobility placement (SMP) program rather than the SM Erasmus exchange one because...

EDIT

The question asks:

Why have you chosen this type of mobility?

Therefore the OP SHOULD reply using the present perfect:

I have chosen the student mobility placement (SMP) program because...

It is unnecessary to add rather than XXX because the question is not asking why you preferred one program over another, the OP has already made his decision.

In addition, the auxiliary would is inappropriate in this case. I would prefer to XXX is used to express what you want to do or wish you could do in a polite manner in the present and NOT in the past.

Verbs that follow prefer, love, like, or dislike can take either the gerund or the infinite.
For more information, see the ELU question:When should a verb be followed by a gerund instead of an infinitive?
Hence in the case of prefer either form is acceptable :

I prefer studying
I prefer to study

To avoid repeating the same noun twice in the same sentence use the pronoun, one.

one and ones

We use one (singular) and ones (plural) to avoid unnecessary repetition.

• See those two girls? Helen is the tall one and Jane is the short one.
• Which is your car, the red one or the blue one?

• My trousers are torn.

I need some new ones.

Therefore I would advise the OP to write

I would prefer to participate in the student mobility placement (SMP) program rather than the SM Erasmus exchange one because...

EDIT

The question asks:

Why have you chosen this type of mobility?

Therefore the OP SHOULD reply using the present perfect:

I have chosen the student mobility placement (SMP) program because...

It is unnecessary to add rather than XXX because the question is not asking why you preferred one program over another, the OP has already made his decision.

In addition, the auxiliary would is inappropriate in this case. I would prefer to XXX is used to express what you want to do or wish you could do in a polite manner in the present and NOT in the past.

Verbs that follow prefer, love, like, or dislike can take either the gerund or the infinite.
For more information, see the ELU question:When should a verb be followed by a gerund instead of an infinitive?
Hence in the case of prefer either form is acceptable :

I would prefer studying
I would prefer to study

To avoid repeating the same noun twice in the same sentence use the pronoun, one.

one and ones

We use one (singular) and ones (plural) to avoid unnecessary repetition.

• See those two girls? Helen is the tall one and Jane is the short one.
• Which is your car, the red one or the blue one?

• My trousers are torn.

I need some new ones.

Therefore I would advise the OP to write

I would prefer to participate in the student mobility placement (SMP) program rather than the SM Erasmus exchange one because...

EDIT

The question asks:

Why have you chosen this type of mobility?

Therefore the OP SHOULD reply using the present perfect:

I have chosen the student mobility placement (SMP) program because...

Verbs that follow prefer, love, like, or dislike can take either the gerund or the infinite.
For more information, see the ELU question:When should a verb be followed by a gerund instead of an infinitive?
Hence in the case of prefer either form is acceptable :

I prefer studying
I prefer to study

To avoid repeating the same noun twice in the same sentence use the pronoun, one.

one and ones

We use one (singular) and ones (plural) to avoid unnecessary repetition.

• See those two girls? Helen is the tall one and Jane is the short one.
• Which is your car, the red one or the blue one?

• My trousers are torn.

I need some new ones.

Therefore I would advise the OP to write

I would prefer to participate in the student mobility placement (SMP) program rather than the SM Erasmus exchange one because...

EDIT

The question asks:

Why have you chosen this type of mobility?

Therefore the OP SHOULD reply using the present perfect:

I have chosen the student mobility placement (SMP) program because...

It is unnecessary to add rather than XXX because the question is not asking why you preferred one program over another, the OP has already made his decision.

In addition, the auxiliary would is inappropriate in this case. I would prefer to XXX is used to express what you want to do or wish you could do in a polite manner in the present and NOT in the past.

Verbs that follow prefer, love, like, or dislike can take either the gerund or the infinite.
For more information, see the ELU question:When should a verb be followed by a gerund instead of an infinitive?
Hence in the case of prefer either form is acceptable :

I would prefer studying
I would prefer to study

To avoid repeating the same noun twice in the same sentence use the pronoun, one.

one and ones

We use one (singular) and ones (plural) to avoid unnecessary repetition.

• See those two girls? Helen is the tall one and Jane is the short one.
• Which is your car, the red one or the blue one?

• My trousers are torn.

I need some new ones.

Therefore I would advise the OP to write

I would prefer to participate in the student mobility placement (SMP) program rather than the SM Erasmus exchange one because...

Verbs that follow prefer, love, like, or dislike can take either the gerund or the infinite.
For more information, see the ELU question:When should a verb be followed by a gerund instead of an infinitive?
Hence in the case of prefer either form is acceptable :

I would prefer studying
I would prefer to study

To avoid repeating the same noun twice in the same sentence use the pronoun, one.

one and ones

We use one (singular) and ones (plural) to avoid unnecessary repetition.

• See those two girls? Helen is the tall one and Jane is the short one.
• Which is your car, the red one or the blue one?

• My trousers are torn.

I need some new ones.

Therefore I would advise the OP to write

I would prefer to participate in the student mobility placement (SMP) program rather than the SM Erasmus exchange one because...

EDIT

The question asks:

Why have you chosen this type of mobility?

Therefore the OP SHOULD reply using the present perfect:

I have chosen the student mobility placement (SMP) program because...