I read in Linga Purana (Poorva bhaga- Chapter 58 that, Lord Brahma nominated Vaasuki as Adhipati of Nagas and Takshaka as Adhipati of Sarpas. What is the difference between Naaga and Sarpa ?
Some differences and similarities are mentioned in Śrīmad-Bhāgavatam (Bhāgavata Purāṇa) » Skandha 5» Adhyay 24 (SB 5.24):
ततोऽधस्तान्महातले काद्रवेयाणां सर्पाणां नैकशिरसां क्रोधवशो नाम गणः कुहक
तक्षककालियसुषेणादिप्रधाना महाभोगवन्तः पतत्त्रिराजाधिपतेः पुरुष
वाहादनवरतमुद्विजमानाः स्वकलत्रापत्यसुहृत्कुटुम्बसङ्गेन क्वचित्प्रमत्ता विहरन्ति ॥ ५.२४.२९ ॥
The planetary system below Talātala is known as Mahātala. It is the abode of many-hooded snakes, descendants of Kadrū, who are always very angry. The great snakes who are prominent are Kuhaka, Takṣaka, Kāliya and Suṣeṇa. The snakes in Mahātala are always disturbed by fear of Garuḍa, the carrier of Lord Viṣṇu, but although they are full of anxiety, some of them nevertheless sport with their wives, children, friends and relatives.
ततोऽधस्तात्पाताले नागलोकपतयो वासुकिप्रमुखाः शङ्खकुलिकमहाशङ्खश्वेत
धनञ्जयधृतराष्ट्रशङ्खचूडकम्बलाश्वतरदेवदत्दयो महाभोगिनो महामर्षा निवसन्ति ।
येषामु ह वै पञ्चसप्तदशशतसहस्रशीर्षाणां फणासु विरचिता महामणयो रोचिष्णवः पाताल विवरतिमिरनिकरं स्वरोचिषा विधमन्ति ॥ ५.२४.३१ ॥
Beneath Rasātala is another planetary system, known as Pātāla or Nāgaloka, where there are many demoniac serpents, the masters of Nāgaloka, such as Śaṅkha, Kulika, Mahāśaṅkha, Śveta, Dhanañjaya, Dhṛtarāṣṭra, Śaṅkhacūḍa, Kambala, Aśvatara and Devadatta. The chief among them is Vāsuki. They are all extremely angry, and they have many, many hoods — some snakes five hoods, some seven, some ten, others a hundred and others a thousand. These hoods are bedecked with valuable gems, and the light emanating from the gems illuminates the entire planetary system of bila-svarga.
Here Vasui is mentioned as king of Nagas, however, in BG 10.28-29, Lord Krishna says:
I am Vasuki among Sarpas
And, I am Ananta among Nagas
That means Vasuki is mentioned as Sarpa in BG 10.28.
Also, Kaliya is mentioned as Sarpa here but in SB 10.16 both Sarpa and Naga words are used for Kaliya.
As both Sarpas and Nagas have same category so sometimes words are used interchangeably for them.
Some differences in them are:
Nagas live in Nagaloka or Patala loka but Sarpas in Mahātala loka
Nagas have gems in their hoods
Nagas have many many many hoods
So with respect to your question, Lord Brahma appointed one as a king of Mahātala loka and other as a king of Nagaloka or Patala loka.