When should I use them, should I use them at all?
Probably never, unless you're writing historical fiction. Archaic and obsolete words are words that are no longer used in contemporary society, so unless you want to specifically emulate olden times, it's best just to leave them alone.
What's the difference between these descriptions?
According to the Standard English section of the M-W preface, archaic words are older, perhaps at least a century out-of-date and used only for a deliberately old-fashioned effect in modern times, while dated words went out of style more recently. Historical words are words that are still used, but only to refer to ancient things. Rare words are words that are slowly leaving the English language.
I don't see obsolete as a usage marker in my edition of M-W, but I would roughly equate it to being between archaic and dated. Obviously, your dictionary's preface should explain how obsolete would be used by their editors.
Also, should I spend time to remember these archaic and obsolete meanings?
Probably not, unless you're reading a lot of historical stuff. You can always look them up in a dictionary if you're only going to encounter them once in a while.
From the OED, Camelopard was first recorded in
▸ a1398 J. Trevisa tr. Bartholomaeus Anglicus De Proprietatibus Rerum (BL Add. 27944) (1975) II. xviii. xx. 1159 Cameleopardus hatte cameleopardalis also... And haþ þe heed of a camele..and spekkes of þe parde. [And has the head of a camel and the spots of the leopard.]
Somewhere between then and 1594 we have early records of "Giraffe":
1594 T. Blundeville Exercises v. ix. f. 259 This beast is called of the Arabians, Gyraffa.
1617 F. Moryson Itinerary i. iii. v. 263 Another beast newly brought out of Affricke..is called..Giraffa by the Italians.
β.
c1600 Sanderson in Purchas Pilgrims (1625) ii. 1619 The admirablest and fairest beast that euer I saw, was a Iarraff.
[Although there was obviously some confusion as the animal was so rare in the UK:
1688 R. Holme Acad. Armory ii. 130/2 Beasts..Such as chew the Cud, and are not Horned, as Camelopard Giraffa.]
It seems from the Google Ngram links above, that the name started to change decisively from Camelopard around the end of the 18th century and the move to Giraffe became all but complete around 1824. This was probably a cultural change that followed the scientific advances in which "Camelopard" was seen as a crude and primitive combination of exterior features, whereas "modern people" required something a little more specific, and hence the Arabic.
Thus we have in The Kaleidoscope: or, Literary and scientific mirror, Volume 8 1821 p81
The ROMANS, to whom Julius Caesar was the first who introduced a giraffe, called the animal camelo-pardalis or camelopard. But a multitude of essential differences distinguish it from the camel; and of the leopard, […] it must also be observed that they [the spots] differ in being flat and irregular, instead of being round and ranged en rose. The ancient name of Zerapha, corrupted by us into giraffe, is much more fitting, therefore, that that which the Romans substituted for it.
As far as other names are concerned, there was apparently one:
1605 J. Sylvester tr. G. de S. Du Bartas Deuine Weekes & Wks. i. vi. 194 Th' horned Hirable [1605 marg. alias, Girafle, 1608 marg. Alias, Gyrafa]
But this may be a mistake of the translator, hence the marginal notes.
Best Answer
These texts are from Middle English and most of the letters are composed of ııııııııııs.
In Middle English, several letters such as u ~ v, i, w, m, n etc., were written using a sequence of a particular short downstroke of the pen/quill, called a minim. A dotless i was a single minim:
So the word 'minim' itself would've been written:
ıııııııııı
It would be ten undifferentiated minims and would've been extremely confusing.
Purveyance
The first word in the question that looks like 'putueaunce' is purueaunce, which was a variant spelling for purveyance. There were identical minims for u and v, so they've been used interchangeably.
A Concise Dictionary of Middle English (Gutenberg) confirms purveaunce and also lists some other variants including purveyaunce, purveiance and purueance.
Or it could be porveaunce which was the Anglo-Norman spelling for purveyance (as pointed out by Andrew Leach and Greybeard in the comments beneath the question).
According to The Language of Emare - A Middle English Romance:
There are lots of quotations for purueaunce/purveiaunce/porveaunce in the Middle English Compendium having many variants of the same spelling such as purveiaunce, purveianse, purveaunce, perveaunce, perviaunce pourveiaunce etc.
Dictionary of the Scots Language also has some quotes using different spellings for purveyance, some of which are purveance, purueianse, purueance, purvyaunce, pourveance, pourueaunce etc.
Here's a quote from Sebastian Brant's The Ship of Fools (vol.2) where purueaunce can be found:
Utility
The second word is utility (as pointed out by Kate Bunting). There were identical minims for u and v. Some sources such as the Oxford English Dictionary claim that they were not exactly the same but were graphic variants of the same letter. The Oxford English Dictionary says that v was used at the beginning of a word and u elsewhere.
The first letter in the second word is therefore a v.
Furthermore, Middle English scribes adopted a practice of replacing an i with a y. Take for example the word mine, it was min in Middle English, so it would've been written ıııııı and would've been indistinguishable from other combinations of relevant letters such as win, nim, nun etc. It was therefore replaced by a y (e.g. myn for 'mine').
The National Archive concurs that y was used for i, and u and v were interchangeable:
From Harvard University Website:
There are many citations for utility/utilité/vtylyte in the Middle English Compendium. Some of the spelling variants they've used for utility are vtylyte, vtilite, utilite etc.
Merriam Webster confirms the spelling of utility with an e:
The spelling vtylyte can also be found in the Dictionary of the Scots Language
Here's a quote from Sebastian Brant's The Ship of Fools (vol. 2) where utility is indeed written as vtylyte: