Sovereignty is a slippery term that has evolved over the centuries. The modern meaning traces back to the end of the Thirty Years War and the Peace of Westphalia known as Westphalian sovereignty, although even this is disputed.
The supreme, absolute, and uncontrollable power by which an independent state is governed and from which all specific political powers are derived; the intentional independence of a state, combined with the right and power of regulating its internal affairs without foreign interference.
Basically it says the sovereign is the "supreme authority within a territory". Nobody else can come in and meddle in your sovereign affairs.
Without a recognition of sovereignty it's unclear who is in charge of what and where. Without territorial sovereignty the ruler of Hereistan might decide they can tell the people in neighboring Theresburg what to do; Theresburg probably won't like that and might have to go to war to stop them. If Hereistan recognizes the sovereignty of neighboring Theresburg they agree not to do that; that would be a violation of Theresburgian sovereignty. Hereistan has to get permission from Theresburg to act within Theresburg.
Of course this still happens all the time, but it's recognized to be wrong.
"Independence" is even more slippery. It usually means internal and external sovereignty; the sovereign has sole control over all affairs within its territory, there is no higher authority.
The status of a fully independent state should be contrasted with that of dependent or vassal states, where a superior state has the legal authority to impose its will over the subject, or inferior, state.
One can have sovereignty over only certain things. Sovereignty can be very roughly split into internal and external sovereignty.
Internal sovereignty means authority over your own internal affairs, but not your relationship with other sovereign states. For example, US states have sovereignty over most things which lie wholly within their borders; other US states cannot interfere, nor can the Federal government (very over-simplified).
External sovereignty is authority over your relationship with other sovereign states. US states do not have external sovereignty; they are not allowed to make treaties with foreign countries. That power is reserved by a superior; the US Federal government.
The US is independent, no other state has legal authority to tell it what to do. US states are not, the US Federal government has legal authority in certain affairs.
Note that voluntarily entering into a treaty with another sovereign state does not violate sovereignty. The sovereign state gives the treaty authority within its borders. Note that things we call "treaties" negotiated by force such as unequal treaties or those imposed upon a defeated nation do violate sovereignty and independence.
The Declaration of State Sovereignty was made on July 27th, 1990, but it did not yet carry the full force of constitutional law until August 1991. I don't see any particular reason to call one "sovereignty" and the other "independence", it all seems to be about independence.
Let's go through the four major relevant events in Belarus's sovereignty/independence, then you can decide.
- Declaration of independence by the Belarus Supreme Soviet.
- Elevation of the declaration by the Belarus Supreme Court to Constitutional law.
- Official recognition of independence by the Soviet Union (by dissolving).
- Adoption of a new constitution.
July 27th, 1990 - Declaration
The Declaration of State Sovereignty of the Belarus SSR is adopted by the Supreme Soviet of Belarus SSR and renames their nation to the Republic of Belarus. This was a full declaration of sovereignty and independence.
...hereby solemnly proclaims the full state sovereignty of the Republic of Belarus as the supremacy, the independence, and the absolute state power of the Republic within its territory, the competence of its laws, the independence of the Republic in foreign relations, and declares its determination to establish a state, based on law.
Article 7 (1) Within the territory of the Republic of Belarus, the Constitution and the laws shall have supremacy.
Article 9 (1) The Republic of Belarus shall be independent in deciding on the questions of culture and spiritual development of the Belarusian nation, other national communities of the Republic, and in organizing its own system of information, education, and upbringing.
Article 11 (1) The Republic of Belarus shall independently exercise the right to enter into voluntary unions with other states and to withdraw freely from these unions.
However, this is still all happening within the framework of the 1978 constitution (which I don't have an English copy of, so my knowledge here will be spotty). The declaration cannot yet override the existing constitution.
25 August, 1991 - Elevation
Following a coup attempt by Soviet hard liners in Moscow, the Supreme Court of the Republic of Belarus gives the Declaration the constitutional law status necessary for it to be acted upon. The Soviet Union does not react. Belarus now has de-facto independence.
8 December, 1991 - Recognition
The Belovezha Accords are signed, the Soviet Union ceases to exist and is replaced with the Commonwealth of Independent States. Belarus's independence from the Soviet Union is now de-jure.
15 March, 1994 - New Constitution
As required by Article 12 of the Declaration...
The provisions of the present Declaration shall be implemented by the Supreme Soviet of the Republic of Belarus through the adoption of a new Constitution (Fundamental Law) of the Republic of Belarus, and laws of the Republic of Belarus.
...Belarus enacts a new constitution to replace its existing one from 1978.
Article 3 On the day on which the Constitution enters into force, the articles of the 1978 Constitution, together with any subsequent amendments and addenda thereto, shall cease to apply, unless otherwise specified in this Law, as shall the articles of the Law on granting the status of a constitutional law to the Declaration of the Supreme Soviet of the Republic of Belarus on the State Sovereignty of the Republic of Belarus of 25 Aug 1991.