There's no such thing as "counting with a bracha" and "counting without a bracha". The Mitzva is to count.
Blessings on (just about) all Mitzvot are separate rabbinic obligations. Whether or not a blessing is said on a Mitzva is a separate question from if there is a Mitzva (some Mitzvot never have blessings!). A Mitzva done without its requisite blessing (if there is one) is still completely effective. As such we generally only say the blessing when we know for sure we are fulfilling the Mitzva, because there is "little" to gain by saying it, and much to lose (Berakha Levatalla -- blessings in vain).
So now just learn the laws of when one is obligated to count the Omer (or whatever other Mitzva you are interested in). If you are for sure obligated, include the blessing. If not, better to omit the blessing.
Some highlights of such a study would include: Most Rishonim assume that one is obligated to count that day's Omer at some point over the 24 hours of each day between Pesach and Shavuot. There are a few Rishonim (eg. Tosfot Menachot 66a) who hold that the Mitzva only applies at night because that's when the barley for the Omer offering was reaped. There is a variant text of the Behag (quoted as a "wonder" in that Tosfot) which says if you miss a day then the obligation ends.
Ok. So if in accordance with most opinions you are counting during the daytime or after you missed a day, better (at least according to the Shulchan Arukh OC 589) to just omit the blessing because it's possible according to certain minority opinions that you aren't fulfilling the Mitzva. Similarly, if you don't remember if you counted already that day, go ahead and count, but leave out the blessing. (Ideally, of course, you'd avoid situations of doubt and fulfill the Mitzva according to everyone by counting every day and always at night.)
The Terumat HaDeshen (#37) writes that if you aren't sure if you missed a day, then you don't need to worry about the opinion of the Behag, since it's only a doubt about a minority opinion and we don't need to be that worried about Berakha Levatalla. This is the ruling of the Shulchan Arukh (OC 589:8). Counting one day during the daytime (or any other doubtful counting) is no worse than being unsure if one counted at all (Levush, etc., ibid.) and one should proceed to further days as if they are for sure obligated to count.