This is a follow-up question of What is an "Ablaut"?, and I was going to answer this in a comment there, but thought asking a new question and answering that properly might be the better way to go.
These two words cause a lot of confusion, which is mainly due to the fact that the difference between the two only becomes clear when you look at language history.
An Ablaut is the systematic change of vowels to form the different tenses. It used to be very systematic in Indoeuropean and the only way to form the past tense, but in the Germanic languages, a new way of expressing this grammatical information emerged which led to the gradual degradation of this older set of rules. If you look at grammars of Middle High German for example, you will find seven Ablautreihen, but they already contain lots of exceptions. This is why sing-sang-sung are Ablauts.
Side note: The new rule is a "d/t-past" and one of the things that sets the Germanic languages apart from the other Indoeuropean languages. Almost all of those "regular verbs" like fill - filled - filled are newer Germanic inventions, except for the cases were a verb lost its "irregularities" and became a "d/t-past" verb. This is also why the related verbs in German are always also "d/t-past" verbs (*füllen - füllte - gefüllt), except when it used to be an irregular verb and assimilated to different degrees.
Now for Umlaut. One meaning of the word are the two dots above a, o, u. But this meaning is secondary (i.e. came later). The primary meaning was the phenomenon that due to assimilation processes an a, o, u in the stem of words sound more like ä, ö, ü in certain cases. Nowadays it is not easy to see why it should be Macht but Mächte, although we also have Päderast and Päderasten where the schwa doesn't seem to induce an assimilation process. But looking at it from an etymological perspective will reveal that the plural of maht used to be mahti in Old High German. The i is a common plural marker in Old High German and also a very high vowel. So it is easier to say mähti, because the pronunciation places of e and i are closer to each other.
The next step in language development was Nebensilbenabschwächung which means that all vowels that didn't have a stress became a schwa. That is why we there is no i any more at the end of Mächte. But the assimilation had already taken place and now also sort of carries the information "there used to be an i here, so it's plural", hence the ä is also written down.
Since now the fact that there is an Umlaut means this is plural and is perceived as such (before the Nebensilbenabschwächung it was just a byproduct), it was then used as a morphological means to form the plural. For example Vater -> Väter never used to be vater -> vatir or something like this.
I'm not too sure about English, but I'll add this because this also came up in the comments to the original question. I suppose that an analogous process took place for foot. Probably in Old English, feet used to be footy or something like that which assimilated to feety and consequently lost the y at the end, so this would be an Umlaut, although it doesn't have "dots" like in German.
I only used the plural as an example here, the other uses of Umlaut (e.g. to form the comparative form of an adjective) emerged in a similar way.
Reference: This is a summary and explanation of the relevant entries in Lexikon der Sprachwissenschaft by Hadumod Bußmann.