I don't think much is known about the early stages of Egyptian writing. We get either a few objects with Hieroglyphic labels in the first few centuries or full blown passages in the pyramid texts a few centuries later on. There is nothing in between to show how the script arose and how it developed into mature writing.
We also assume that the hieroglyphs we normally see and use were actually a ceremonial elaborations of a more basic and simpler script closer in form to what we call Hieratic. Even so, the iconography glyphs used in the Sumerian and Egyptian scripts have nothing in common beyond chance. E.g., Sumerian "house" (/e/) is 𒂍, and Egyptian "house" (/pr/) is 
. Aside from this difference, it is hard to compare the earliest stages of the scripts because of this lack of early Egyptian writing.
What I say below refers to what we see in the earliest Egyptian sentences. I am also obliged to use a later form of the Cuneiform glyphs used for writing in clay with a reed stylus, rather than the earlier more picture like glyphs.
Perhaps, because of the difference in language phonology, Sumerian writes with syllabic glyphs that always include a vowel, whereas Egyptian tries to write with glyphs representing 1-3 consonants, irrespective of the location or presence of the vowels in the spoken language.
Both scripts often use phonetic writing instead of an iconic glyph, but Sumerian also uses phonetic writing to indicate the last syllable of the stem plus a following particle beginning with a vowel. Egyptian never does this. Egyptian quite often indicates the consonants redundantly, using both unilateral signs and bilateral or trilateral signs. E.g., in "xnm" 
, is written phonetically as "x-n nm m" plus two determinatives.
A fourth difference is that some determinatives in Sumerian precede the rest of the word (e.g., 𒀭 "an" as a determinative of a God can precede the God "Enlil" 𒀭𒂗𒆤) and some follow (e.g., 𒆠 "ki" as a determinative of a city/country folows "Uruk"/"Unug" 𒀕𒆠). In Egyptian, determinatives only follow they accompanying phonetic glyphs.
A fifth difference that may have only developed later is that I think Sumerian typically uses only one determinative, whereas Egyptian often uses several. E.g.:
The two figures of the people at the end are both determinatives. Even when hieroglyphs are used iconically, they are usually used with a | underneath to show this, as in the glyph for "pr" above. Most substantive Egyptian words use at least one determinative, almost as word dividers; whereas I think determinatives are much more sparingly used in Sumerian writing.
A sixth difference that may have developed later is that many Egyptian determinatives have no other use. If you look at Gardiner's List under A, I can only recall a handful of these being used other than as a determinatives. Notice how many have so many "readings" that they are basically useless in predicting what word they are used with (e.g., A2 and A19).
A seventh difference is that Cuneiform glyphs sometimes combine in ways that Egyptian glyphs do not. E.g., 𒈗 (lugal, "king") is a combination of 𒇽 (lu, "man") and 𒃲 (gal, "great/large cup") written together. You can't do that with hieroglyphs.
With early Egyptian glyphs, duplication was used to indicate a noun with a dual ending, and writing it three times was used to show the plural (which usually had a different vowel ending and perhaps some internal changes). In cuneiform, repeated glyphs show a repeated syllable.
It is possible that earlier Egyptian scribes were aware of Sumerian writing, but it is highly doubtful they had any skill in writing or reading it. Old Egyptian and Akkadian are distantly related languages and broadly similar in morphology, but their scripts have nothing in common that they do not also share with Mayan and Old Chinese. We see how scribes knowledgeable in Sumerian script adapted it for writing Akkadian, and we have many examples of other scripts that were adapted to write unrelated languages (e.g., Hittite, Japanese, Tibetan). Egyptian shows no hint of such a process of adaptation.
